1. Show that if \(A^3 = 4I_n\text{,}\) then \(A\) is invertible.

2. Show that if \(A^2-3A+2I_n=0\text{,}\) then \(A\) is invertible.

3. Simplify the expression \(B(AB)^{-1}(AB)^2B^{-1}\text{.}\)

4. Find the inverse of \(A = \bbm 1\amp 4\\2\amp 5\ebm\text{.}\)

Let \(A\) and \(B\) be invertible matrices.

1. The inverse of \(A\) is unique.

2. \(A^{-1}\) is also invertible, and \((A^{-1})^{-1}=A\text{.}\)

3. \(AB\) is invertible, and \((AB)^{-1}=B^{-1}A^{-1}\text{.}\)

4. For any nonzero scalar \(k\text{,}\) \(kA\) is invertible, and \((kA)^{-1}=\frac1k A^{-1}\text{.}\)

5. For any \(n\times 1\) vector \(\vec{b}\text{,}\) the unique solution to \(A\vec{x}=\vec{b}\) is \(\vec{x}=A^{-1}\vec{b}\text{.}\)

Recall: if \(A\) is an \(m\times n\) matrix, and \(\vec{x}\) is an \(n\times 1\) vector, then \(\vec{y}=A\vec{x}\) is an \(m\times 1\) vector.

Let \(\R^n\) (sometimes, \(\R^{n,1}\)) denote the set of all \(n\times 1\) column vectors.

Then any \(m\times n\) matrix \(A\) can be used to define a function

What properties does this function have?

Let \(A = \bbm 3\amp -2\\5\amp 4\ebm\text{,}\) and let \(T(\vec{x})=A\vec{x}\text{.}\)

Compute the value of \(T\) on:

Repeat the above for \(A = \bbm 1\amp -2\\-2\amp 4\ebm\text{.}\)

Note: any function between vector spaces (e.g. \(\R^n,\R^m\)) with these properties is called a linear transformation.

It turns out any linear transformation can be expressed as a matrix transformation.

1. Show that for any linear transformation, \(T(\vec{0})=\vec{0}\text{.}\)

2. 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 \(T\left(\bbm 2\\3\ebm\right)\)

3. 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 \(T(2\vec{a}-3\vec{b}+5\vec{c})\text{.}\)

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 \(T(\vec{x})=A\vec{x}\text{?}\)

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.