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Linear Algebra for Team-Based Inquiry Learning: 2023 Edition

Steven Clontz, Drew Lewis

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Section 3.3 Image and Kernel (AT3)

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Subsection 3.3.1 Class Activities

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Activity 3.3.1.

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Let T:R2→R3 be given by
T([xy])=[xy0]with standard matrix [100100]
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Which of these subspaces of R2 describes the set of all vectors that transform into 0→?
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    {[aa]|a∈R}
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    {[a0]|a∈R}
  3. 🔗
    {[00]}
  4. 🔗
    {[ab]|a,b∈R}
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Definition 3.3.2.

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Let T:V→W be a linear transformation, and let z→ be the additive identity (the “zero vector”) of W. The kernel of T is an important subspace of V defined by
ker⁡T={v→∈V | T(v→)=z→}
Figure 24. The kernel of a linear transformation
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Activity 3.3.3.

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Let T:R3→R2 be given by
T([xyz])=[xy]with standard matrix [100010]
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Which of these subspaces of R3 describes ker⁡T, the set of all vectors that transform into 0→?
  1. 🔗
    {[00a]|a∈R}
  2. 🔗
    {[aa0]|a∈R}
  3. 🔗
    {[000]}
  4. 🔗
    {[abc]|a,b,c∈R}
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Activity 3.3.4.

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Let T:R3→R2 be the linear transformation given by the standard matrix
T([xyz])=[3x+4y−zx+2y+z]
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(a)
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Set T([xyz])=[00] to find a linear system of equations whose solution set is the kernel.
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(b)
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Use RREF(A) to solve this homogeneous system of equations and find a basis for the kernel of T.
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Activity 3.3.5.

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Let T:R4→R3 be the linear transformation given by
T([xyzw])=[2x+4y+2z−4w−2x−4y+z+w3x+6y−z−4w].
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Find a basis for the kernel of T.
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Activity 3.3.6.

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Let T:R2→R3 be given by
T([xy])=[xy0]with standard matrix [100100]
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Which of these subspaces of R3 describes the set of all vectors that are the result of using T to transform R2 vectors?
  1. 🔗
    {[00a]|a∈R}
  2. 🔗
    {[ab0]|a,b∈R}
  3. 🔗
    {[000]}
  4. 🔗
    {[abc]|a,b,c∈R}
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Definition 3.3.7.

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Let T:V→W be a linear transformation. The image of T is an important subspace of W defined by
ImT={w→∈W | there is some v→∈V with T(v→)=w→}
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In the examples below, the left example’s image is all of R2, but the right example’s image is a planar subspace of R3.
Figure 25. The image of a linear transformation
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Activity 3.3.8.

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Let T:R3→R2 be given by
T([xyz])=[xy]with standard matrix [100010]
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Which of these subspaces of R2 describes ImT, the set of all vectors that are the result of using T to transform R3 vectors?
  1. 🔗
    {[aa]|a∈R}
  2. 🔗
    {[a0]|a∈R}
  3. 🔗
    {[00]}
  4. 🔗
    {[ab]|a,b∈R}
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Activity 3.3.9.

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Let T:R4→R3 be the linear transformation given by the standard matrix
A=[3471−1102213−1]=[T(e→1)T(e→2)T(e→3)T(e→4)].
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Since for a vector v→=[x1x2x3x4], T(v→)=T(x1e→1+x2e→2+x3e→3+x4e→4), which of the following best describes the set of vectors
{[3−12],[411],[703],[12−1]}?
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    The set of vectors spans ImT but is not linearly independent.
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    The set of vectors is a linearly independent subset of ImT but does not span ImT.
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    The set of vectors is linearly independent and spans ImT; that is, the set of vectors is a basis for ImT.
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Observation 3.3.10.

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Let T:R4→R3 be the linear transformation given by the standard matrix
A=[3471−1102213−1].
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Since the set {[3−12],[411],[703],[12−1]} spans ImT, we can obtain a basis for ImT by finding RREFA=[101−101110000] and only using the vectors corresponding to pivot columns:
{[3−12],[411]}
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Activity 3.3.12.

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Let T:R3→R4 be the linear transformation given by the standard matrix
A=[1−322−60001−131].
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Find a basis for the kernel and a basis for the image of T.
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Activity 3.3.13.

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Let T:Rn→Rm be a linear transformation with standard matrix A. Which of the following is equal to the dimension of the kernel of T?
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    The number of pivot columns
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    The number of non-pivot columns
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    The number of pivot rows
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    The number of non-pivot rows
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Activity 3.3.14.

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Let T:Rn→Rm be a linear transformation with standard matrix A. Which of the following is equal to the dimension of the image of T?
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    The number of pivot columns
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    The number of non-pivot columns
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    The number of pivot rows
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    The number of non-pivot rows
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Observation 3.3.15.

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Combining these with the observation that the number of columns is the dimension of the domain of T, we have the rank-nullity theorem:
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The dimension of the domain of T equals dim⁡(ker⁡T)+dim⁡(ImT).
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The dimension of the image is called the rank of T (or A) and the dimension of the kernel is called the nullity.
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Activity 3.3.16.

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Let T:R4→R3 be the linear transformation given by
T([xyzw])=[x−y+5z+3w−x−4z−2wy−2z−w].
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(a)
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Explain and demonstrate how to find the image of T and a basis for that image.
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(b)
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Explain and demonstrate how to find the kernel of T and a basis for that kernel.
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(c)
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Explain and demonstrate how to find the rank and nullity of T, and why the rank-nullity theorem holds for T.
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Subsection 3.3.2 Videos

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Figure 26. Video: The kernel and image of a linear transformation
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Figure 27. Video: Finding a basis of the image of a linear transformation
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Figure 28. Video: Finding a basis of the kernel of a linear transformation
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Figure 29. Video: The rank-nullity theorem
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Subsection 3.3.3 Slideshow

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Exercises 3.3.4 Exercises

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Subsection 3.3.5 Mathematical Writing Explorations

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Exploration 3.3.17.

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Assume f:V→W is a linear map. Let {v1→,v2→,…,vn→} be a set of vectors in V, and set wi→=f(vi→).
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    If the set {w1→,w2→,…,wn→} is linearly independent, must the set {v1→,v2→,…,vn→} also be linearly independent?
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    If the set {v1→,v2→,…,vn→} is linearly independent, must the set {w1→,w2→,…,wn→} also be linearly independent?
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    If the set {w1→,w2→,…,wn→} spans W, must the set {v1→,v2→,…,vn→} also span V?
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    If the set {v1→,v2→,…,vn→} spans V, must the set {w1→,w2→,…,wn→} also span W?
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    In light of this, is the image of the basis of a vector space always a basis for the codomain?
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Exploration 3.3.18.

Prove the Rank-Nullity Theorem. Use the steps below to help you.
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    The theorem states that, given a linear map h:V→W, with V and W vector spaces, the rank of h, plus the nullity of h, equals the dimension of the domain V. Assume that the dimension of V is n.
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    For simplicity, denote the rank of h by R(h), and the nullity by N(h).
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    Recall that R(h) is the dimension of the range space of h. State the precise definition.
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    Recall that N(h) is the dimension of the null space of h. State the precise definition.
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    Begin with a basis for the null space, denoted BN={β1→,β2→,…,βk→}. Show how this can be extended to a basis BV for V, with BV={β1→,β2→,…,βk→,βk+1→,βk+2→,…,βn→}. In this portion, you should assume k≤n, and construct additional vectors which are not linear combinations of vectors in BN. Prove that you can always do this until you have n total linearly independent vectors.
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    Show that BR={h(βk+1→),h(βk+2→),…,h(βn→)} is a basis for the range space. Start by showing that it is linearly independent, and be sure you prove that each element of the range space can be written as a linear combination of BR.
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    Show that BR spans the range space.
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    State your conclusion.
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Subsection 3.3.6 Sample Problem and Solution

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Sample problem Example B.1.14.
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