Assume \(T\) is a linear transformation.

1. Given \(T\left(\bbm 1\\0\ebm\right) = \bbm 3\\-2\ebm\) and \(T\left(\bbm 0\\1\ebm\right) = \bbm -4\\5\ebm\text{,}\) find \(T\left(\bbm 2\\3\ebm\right)\)

2. Given \(T(\vec{a})=\bbm 1\\0\\-3\ebm\text{,}\) \(T(\vec{b})=\bbm 0\\-2\\5\ebm\) and \(T(\vec{c})=\bbm 1\\1\\1\ebm\text{,}\) find \(T(2\vec{a}-3\vec{b}+5\vec{c})\text{.}\)

Reminder: \(T\) is linear if \(T(\vec{x}+\vec{y})=T(\vec{x})+T(\vec{y})\) and \(T(k\vec{x})=kT(\vec{x})\) for any vectors \(\vec{x}, \vec{y}\) and scalar \(k\text{.}\)

For each map \(T:\R^2\to\R^2\text{,}\) decide if it's linear:

1. \(T\left(\bbm x\\y\ebm\right) = \bbm 2x-3y\\x+4y\ebm\)

2. \(T\left(\bbm x\\y\ebm\right) = \bbm 4xy\\2x+y\ebm\)

3. \(T\left(\bbm x\\y\ebm\right) = \bbm 2x+1\\x-3y+2\ebm\)

1. If \(T(\vec{x})=A\vec{x}\) for \(A = \bbm 2\amp -3\\-1\amp -2\\5\amp 4\ebm\text{,}\) what are the domain and codomain of \(T\text{?}\)

2. For \(T\) as above, compute \(T(\hat{\imath})\) and \(T(\hat{\jmath})\)

3. What if \(A = \bbm 2\amp -1\amp 5\\4\amp -2\amp 3\ebm\text{?}\)

4. If \(T\left(\bbm x\\y\\z\ebm\right) = \bbm 3x-2y\\-x+4y+5z\\7x-2y-6z\ebm\text{,}\) for what matrix is \(T(\vec{x})=A\vec{x}\text{?}\)

1. Determine the matrix of \(T:\R^3\to \R^2\text{,}\) given that

   \begin{equation*} T(\hat{\imath}) = \bbm 2\\-1\ebm, T(\hat{\jmath}) = \bbm -3\\5\ebm, T(\hat{k}) = \bbm -7\\6\ebm\text{.} \end{equation*}

2. Determine the matrix of \(T:\R^2\to \R^2\text{,}\) given that

   \begin{equation*} T\left(\bbm 2\\-1\ebm\right)=\bbm 3\\4\ebm, T\left(\bbm -1\\1\ebm\right) = \bbm -1\\2\ebm\text{.} \end{equation*}

When \(A\) is \(2\times 2\) we can visualize everything in terms of geometric vectors in the plane.

We can use matrices to describe transformations, like stretches, rotations, and reflections.

(But not translations.)

• Stretches: \(\bbm k\amp 0\\0\amp 1\ebm, \bbm 1\amp 0\\0\amp k\ebm, \bbm k\amp 0\\0\amp k\ebm = kI_2\text{.}\) (This is just scalar multiplication.)

• Reflections: \(\bbm -1\amp 0\\0\amp 1\ebm, \bbm 1\amp 0\\0\amp -1\ebm, \bbm -1\amp 0\\0\amp -1\ebm, \bbm 0\amp 1\\1\amp 0\ebm\)

• Rotations: \(\bbm \cos(\theta)\amp -\sin(\theta)\\ \sin(\theta)\amp \cos(\theta)\ebm\)

• Shears: \(\bbm 1\amp k\\0\amp 1\ebm, \bbm 1\amp 0\\k\amp 1\ebm\)

Determine the matrix transformation that:

1. Stretches horizontally by a factor of 2, rotates by \(90^\circ\text{,}\) and then reflects across the \(x\) axis.

2. Reflects across the line \(y=x\text{,}\) stretches vertically be a factor of 3, then reflects across the \(y\) axis.

Let \(T:\R^n\to\R^m\) be a linear map. The range of \(T\) is the set of all \(\vec{y}\in\R^m\) that are equal to \(T(\vec{x})\) for some \(\vec{x}\in \R^n\text{.}\)

In other words: if \(T(\vec{x})=A\vec{x}\text{,}\) the range of \(T\) is the set of all \(\vec{y}\) for which the system \(A\vec{x}=\vec{y}\) is consistent.

Recall: if \(A = \bbm A_1\amp A_2\amp\cdots A_n\ebm\) (in terms of columns) then

So the range of \(T\) is all linear combinations of the columns of \(A\text{.}\) (This is why range is also called column space.)

Determine the range of:

1. \(S(\vec{x})=A\vec{x}\text{,}\) \(A = \bbm 1\amp 2\\3\amp 6\\-2\amp -4\ebm\)

2. \(T(\vec{x})=B\vec{x}\text{,}\) \(B = \bbm 1\amp -3\amp 2\\-1\amp 3\amp 5\ebm\)

The null space of a linear transformation \(T(\vec{x})=A\vec{x}\) is the set of all vectors \(\vec{x}\) such that \(T(\vec{x})=\vec{0}\text{.}\)

In other words, it's the set of all solutions to the homogeneous system \(A\vec{x}=\vec{0}\text{.}\)

Both null space and column space are examples of subspaces.

Given a subspace, we often want to find a basis for it. This is a sort of “minimal generating set” of vectors.

A basis for the null space is the set of basic solutions to \(A\vec{x}=\vec{0}\text{.}\)

Determine the null space and column space of the transformation \(T\) with matrix