Recall that a linear map \(T:V\rightarrow W\) satisfies

1. \(T(\vec{v}+\vec{w}) = T(\vec{v})+T(\vec{w})\) for any \(\vec{v},\vec{w} \in V\text{.}\)

2. \(T(c\vec{v}) = cT(\vec{v})\) for any \(c \in \IR,\vec{v} \in V\text{.}\)

In other words, a map is linear when vector space operations can be applied before or after the transformation without affecting the result.

Suppose \(T: \IR^3 \rightarrow \IR^2\) is a linear map, and you know \(T\left(\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] \right) = \left[\begin{array}{c} 2 \\ 1 \end{array}\right]\) and \(T\left(\left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \right) = \left[\begin{array}{c} -3 \\ 2 \end{array}\right] \text{.}\) What is \(T\left(\left[\begin{array}{c} 3 \\ 0 \\ 0 \end{array}\right]\right)\text{?}\)

1. \(\displaystyle \left[\begin{array}{c} 6 \\ 3\end{array}\right]\)
2. \(\displaystyle \left[\begin{array}{c} -9 \\ 6 \end{array}\right]\)
3. \(\displaystyle \left[\begin{array}{c} -4 \\ -2 \end{array}\right]\)
4. \(\displaystyle \left[\begin{array}{c} 6 \\ -4 \end{array}\right]\)

Suppose \(T: \IR^3 \rightarrow \IR^2\) is a linear map, and you know \(T\left(\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] \right) = \left[\begin{array}{c} 2 \\ 1 \end{array}\right]\) and \(T\left(\left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \right) = \left[\begin{array}{c} -3 \\ 2 \end{array}\right] \text{.}\) What is \(T\left(\left[\begin{array}{c} 1 \\ 0 \\ 1 \end{array}\right]\right)\text{?}\)

1. \(\displaystyle \left[\begin{array}{c} 2 \\ 1\end{array}\right]\)

2. \(\displaystyle \left[\begin{array}{c} 3 \\ -1 \end{array}\right]\)

3. \(\displaystyle \left[\begin{array}{c} -1 \\ 3 \end{array}\right]\)

4. \(\displaystyle \left[\begin{array}{c} 5 \\ -8 \end{array}\right]\)

Suppose \(T: \IR^3 \rightarrow \IR^2\) is a linear map, and you know \(T\left(\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] \right) = \left[\begin{array}{c} 2 \\ 1 \end{array}\right]\) and \(T\left(\left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \right) = \left[\begin{array}{c} -3 \\ 2 \end{array}\right] \text{.}\) What is \(T\left(\left[\begin{array}{c} -2 \\ 0 \\ -3 \end{array}\right]\right)\text{?}\)

1. \(\displaystyle \left[\begin{array}{c} 2 \\ 1\end{array}\right]\)

2. \(\displaystyle \left[\begin{array}{c} 3 \\ -1 \end{array}\right]\)

3. \(\displaystyle \left[\begin{array}{c} -1 \\ 3 \end{array}\right]\)

4. \(\displaystyle \left[\begin{array}{c} 5 \\ -8 \end{array}\right]\)

Suppose \(T: \IR^3 \rightarrow \IR^2\) is a linear map, and you know \(T\left(\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] \right) = \left[\begin{array}{c} 2 \\ 1 \end{array}\right]\) and \(T\left(\left[\begin{array}{c} 0 \\ 0 \\ 1 \end{array}\right] \right) = \left[\begin{array}{c} -3 \\ 2 \end{array}\right] \text{.}\) What piece of information would help you compute \(T\left(\left[\begin{array}{c}0\\4\\-1\end{array}\right]\right)\text{?}\)

1. The value of \(T\left(\left[\begin{array}{c} 0\\-4\\0\end{array}\right]\right)\text{.}\)

2. The value of \(T\left(\left[\begin{array}{c} 0\\1\\0\end{array}\right]\right)\text{.}\)

3. The value of \(T\left(\left[\begin{array}{c} 1\\1\\1\end{array}\right]\right)\text{.}\)

4. Any of the above.

Consider any basis \(\{\vec b_1,\dots,\vec b_n\}\) for \(V\text{.}\) Since every vector \(\vec v\) can be written as a linear combination of basis vectors, \(\vec v = x_1\vec b_1+\dots+ x_n\vec b_n\text{,}\) we may compute \(T(\vec v)\) as follows:

Therefore any linear transformation \(T:V \rightarrow W\) can be defined by just describing the values of \(T(\vec b_i)\text{.}\)

Put another way, the images of the basis vectors completely determine the transformation \(T\text{.}\)

Since a linear transformation \(T:\IR^n\to\IR^m\) is determined by its action on the standard basis \(\{\vec e_1,\dots,\vec e_n\}\text{,}\) it is convenient to store this information in an \(m\times n\) matrix, called the standard matrix of \(T\text{,}\) given by \([T(\vec e_1) \,\cdots\, T(\vec e_n)]\text{.}\)

For example, let \(T: \IR^3 \rightarrow \IR^2\) be the linear map determined by the following values for \(T\) applied to the standard basis of \(\IR^3\text{.}\)

Then the standard matrix corresponding to \(T\) is

Let \(T: \IR^4 \rightarrow \IR^3\) be the linear transformation given by

Let \(T: \IR^3 \rightarrow \IR^2\) be the linear transformation given by

Part 1.

Compute \(T(\vec e_1)\text{,}\) \(T(\vec e_2)\text{,}\) and \(T(\vec e_3)\text{.}\)

Let \(T: \IR^3 \rightarrow \IR^2\) be the linear transformation given by

Part 2.

Find the standard matrix for \(T\text{.}\)

Because every linear map \(T:\IR^m\to\IR^n\) has a linear combination of the variables in each component, and thus \(T(\vec e_i)\) yields exactly the coefficients of \(x_i\text{,}\) the standard matrix for \(T\) is simply an ordered list of the coefficients of the \(x_i\text{:}\)

Let \(T: \IR^3 \rightarrow \IR^3\) be the linear transformation given by the standard matrix

Part 1.

Compute \(T\left(\left[\begin{array}{c} 1\\ 2 \\ 3 \end{array}\right] \right) \text{.}\)

Let \(T: \IR^3 \rightarrow \IR^3\) be the linear transformation given by the standard matrix

Part 2.

Compute \(T\left(\left[\begin{array}{c} x\\ y \\ z \end{array}\right] \right) \text{.}\)

Compute the following linear transformations of vectors given their standard matrices.

Part 1.

Compute the following linear transformations of vectors given their standard matrices.

Part 2.

Compute the following linear transformations of vectors given their standard matrices.

Part 3.