Chapter 10: Linear algebraContent check — Differential Equations (Lebl), Chapter 10: Linear algebra

Compute the magnitude of

a) $\left[\begin{array}{c}1\\ 3\end{array}\right]$

b) $\left[\begin{array}{c}2\\ 3\\ -1\end{array}\right]$

c) $(-\mathrm{2,1},-2)$

Find the unit vector in the direction of the given vector

a) $\left[\begin{array}{c}-1\\ 1\end{array}\right]$

b) $\left[\begin{array}{c}1\\ -1\\ 2\end{array}\right]$

c) $(2,-\mathrm{5,2})$

Compute

a) $\left[\begin{array}{c}3\\ 1\end{array}\right]+\left[\begin{array}{c}6\\ -3\end{array}\right]$

b) $\left[\begin{array}{c}-1\\ 2\end{array}\right]-\left[\begin{array}{c}2\\ -1\end{array}\right]$

c) $-\left[\begin{array}{c}-5\\ 3\end{array}\right]$

d) $2\left[\begin{array}{c}-2\\ 4\end{array}\right]$

e) $3\left[\begin{array}{c}1\\ 0\end{array}\right]+7\left[\begin{array}{c}0\\ 1\end{array}\right]$

f) $2\left[\begin{array}{c}2\\ -3\end{array}\right]-6\left[\begin{array}{c}2\\ -1\end{array}\right]$

If the magnitude of $𝐱$ is 5, what is the magnitude of

a) $4𝐱$

b) $-2𝐱$

c) $-4𝐱$

Suppose a linear mapping $F:{ℝ}^2\to {ℝ}^2$ takes $(\mathrm{1,0})$ to $(1,-1)$ and it takes $(\mathrm{0,1})$ to $(\mathrm{2,0})$⁠. Where does it take

a) $(\mathrm{1,1})$

b) $(\mathrm{0,2})$

c) $(1,-1)$

Add the following matrices

a) $\left[\begin{array}{ccc}2& 1& 0\\ 1& 1& -1\end{array}\right]+\left[\begin{array}{ccc}5& 3& 4\\ 1& 2& 5\end{array}\right]$

b) $\left[\begin{array}{ccc}6& -2& 3\\ 7& 3& 3\\ 8& -1& 2\end{array}\right]+$ $\left[\begin{array}{ccc}-1& -1& -3\\ 6& 7& 3\\ -9& 4& -1\end{array}\right]$

Compute

a) $2\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]+3\left[\begin{array}{cc}-1& 3\\ 1& 2\end{array}\right]$

b) $3\left[\begin{array}{cc}2& -1\\ 1& 3\end{array}\right]-2\left[\begin{array}{cc}2& 1\\ -1& 2\end{array}\right]$

Multiply the following matrices

a) $\left[\begin{array}{ccc}2& 1& 4\\ 3& 4& 4\end{array}\right]\left[\begin{array}{cc}2& 4\\ 6& 3\\ 3& 5\end{array}\right]$

b) $\left[\begin{array}{ccc}0& 3& 3\\ 2& -2& 1\\ 3& 5& -2\end{array}\right]\left[\begin{array}{ccc}6& 6& 2\\ 4& 6& 0\\ 2& 0& 4\end{array}\right]$

c) $\left[\begin{array}{ccc}3& 4& 1\\ 2& -1& 0\\ 4& -1& 5\end{array}\right]\left[\begin{array}{cccc}0& 2& 5& 0\\ 2& 0& 5& 2\\ 3& 6& 1& 6\end{array}\right]$

d) $\left[\begin{array}{cc}-2& -2\\ 5& 3\\ 2& 1\end{array}\right]\left[\begin{array}{cc}0& 3\\ 1& 3\end{array}\right]$

Compute the inverse of the given matrices

a) $\left[\begin{array}{c}2\end{array}\right]$

b) $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

c) $\left[\begin{array}{cc}1& 2\\ 3& 5\end{array}\right]$

d) $\left[\begin{array}{cc}4& 2\\ 4& 4\end{array}\right]$

Compute the inverse of the given matrices

a) $\left[\begin{array}{cc}2& 0\\ 0& 3\end{array}\right]$

b) $\left[\begin{array}{ccc}4& 0& 0\\ 0& 5& 0\\ 0& 0& -1\end{array}\right]$

c) $\left[\begin{array}{cccc}-1& 0& 0& 0\\ 0& 2& 0& 0\\ 0& 0& 3& 0\\ 0& 0& 0& 0.1\end{array}\right]$

Compute the reduced row echelon form for the following matrices:

a) $\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\end{array}\right]$

b) $\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$

c) $\left[\begin{array}{cc}1& 1\\ -2& -2\end{array}\right]$

d) $\left[\begin{array}{ccc}1& -3& 1\\ 4& 6& -2\\ -2& 6& -2\end{array}\right]$

e) $\left[\begin{array}{cccc}2& 2& 5& 2\\ 1& -2& 4& -1\\ 0& 3& 1& -2\end{array}\right]$

f) $\left[\begin{array}{cccc}-2& 6& 4& 3\\ 6& 0& -3& 0\\ 4& 2& -1& 1\end{array}\right]$

g) $\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$

h) $\left[\begin{array}{cccc}1& 2& 3& 3\\ 1& 2& 3& 5\end{array}\right]$

Compute the inverse of the given matrices

a) $\left[\begin{array}{ccc}0& 1& 0\\ -1& 0& 0\\ 0& 0& 1\end{array}\right]$

b) $\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 0\\ 1& 0& 0\end{array}\right]$

c) $\left[\begin{array}{ccc}2& 4& 0\\ 2& 2& 3\\ 2& 4& 1\end{array}\right]$

Solve (find all solutions), or show no solution exists

a) $\begin{array}{cc}\hfill 4x_1+3x_2& =-1\hfill \\ \hfill 5x_1+6x_2& =4\hfill \end{array}$

b) $\begin{array}{cc}\hfill 5x+6y+5z& =7\hfill \\ \hfill 6x+8y+6z& =-1\hfill \\ \hfill 5x+2y+5z& =2\hfill \end{array}$

c) $\begin{array}{cc}\hfill a+\phantom{5}b+\phantom{6}c& =-1\hfill \\ \hfill a+5b+6c& =-1\hfill \\ \hfill -2a+5b+6c& =8\hfill \end{array}$

d) $\begin{array}{cc}\hfill -2x_1+2x_2+8x_3& =6\hfill \\ \hfill x_2+\phantom{8}{x}_3& =2\hfill \\ \hfill x_1+4x_2+\phantom{8}{x}_3& =7\hfill \end{array}$

By computing the inverse, solve the following systems for $𝐱$⁠.

a) $\left[\begin{array}{cc}-1& 1\\ 3& 3\end{array}\right]𝐱=\left[\begin{array}{c}4\\ 6\end{array}\right]$

b) $\left[\begin{array}{cc}2& 7\\ 1& 6\end{array}\right]𝐱=\left[\begin{array}{c}1\\ 3\end{array}\right]$

Compute the rank of the given matrices

a) $\left[\begin{array}{ccc}7& -1& 6\\ 7& 7& 7\\ 7& 6& 2\end{array}\right]$

b) $\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 2& 2& 2\end{array}\right]$

c) $\left[\begin{array}{ccc}0& 3& -1\\ 6& 3& 1\\ 4& 7& -1\end{array}\right]$

For the matrices in Exercise A.3.105, find a linearly independent set of row vectors that span the row space (they do not need to be rows of the matrix).

For the matrices in Exercise A.3.105, find a linearly independent set of columns that span the column space. That is, find the pivot columns of the matrices.

Find a linearly independent subset of the following vectors that has the same span.

$$\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}3\\ 1\\ -5\end{array}\right],\left[\begin{array}{c}0\\ 3\\ -1\end{array}\right],\left[\begin{array}{c}-3\\ 2\\ 4\end{array}\right]$$

For the following sets of vectors, find a basis for the subspace spanned by the vectors, and find the dimension of the subspace.

a) $\left[\begin{array}{c}1\\ 2\end{array}\right],\left[\begin{array}{c}1\\ 1\end{array}\right]$

b) $\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],\left[\begin{array}{c}2\\ 2\\ 2\end{array}\right],\left[\begin{array}{c}1\\ 1\\ 2\end{array}\right]$

c) $\left[\begin{array}{c}5\\ 3\\ 1\end{array}\right],\left[\begin{array}{c}5\\ -1\\ 5\end{array}\right],\left[\begin{array}{c}-1\\ 3\\ -4\end{array}\right]$

d) $\left[\begin{array}{c}2\\ 2\\ 4\end{array}\right],\left[\begin{array}{c}2\\ 2\\ 3\end{array}\right],\left[\begin{array}{c}4\\ 4\\ -3\end{array}\right]$

e) $\left[\begin{array}{c}1\\ 0\end{array}\right],\left[\begin{array}{c}2\\ 0\end{array}\right],\left[\begin{array}{c}3\\ 0\end{array}\right]$

f) $\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}2\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 2\end{array}\right]$

For the following matrices, find a basis for the kernel (nullspace).

a) $\left[\begin{array}{cccc}2& 6& 1& 9\\ 1& 3& 2& 9\\ 3& 9& 0& 9\end{array}\right]$

b) $\left[\begin{array}{ccc}2& -2& -5\\ -1& 1& 5\\ -5& 5& -3\end{array}\right]$

c) $\left[\begin{array}{ccc}1& -5& -4\\ 2& 3& 5\\ -3& 5& 2\end{array}\right]$

d) $\left[\begin{array}{ccc}0& 4& 4\\ 0& 1& 1\\ 0& 5& 5\end{array}\right]$

Suppose the column space of a $9\times 5$ matrix $A$ is of dimension 3. Find

a) Rank of $A$⁠.

b) Nullity of $A$⁠.

c) Dimension of the row space of $A$⁠.

d) Dimension of the nullspace of $A$⁠.

e) Size of the maximum subset of linearly independent rows of $A$⁠.

Find the $s$ that makes the following vectors orthogonal: $(\mathrm{1,1,1})$⁠, $(1,s,1)$⁠.

Find the angle $\theta$ between $(\mathrm{1,2,3})$⁠, $(\mathrm{1,1,1})$⁠.

Given that $⟨𝐯,𝐰⟩=1$⁠, $⟨𝐯,𝐮⟩=-1$⁠, and $\Vert 𝐯\Vert =3$⁠, compute

a) $⟨3𝐮,5𝐯⟩$

b) $⟨𝐯,2𝐰+3𝐮⟩$

c) $⟨𝐰+3𝐯,𝐯⟩$

Suppose $𝐯=(\mathrm{1,0},-1)$⁠. Find

a) ${\mathrm{proj}}_{𝐯}((\mathrm{0,2,1}))$

b) ${\mathrm{proj}}_{𝐯}((\mathrm{1,0,1}))$

c) ${\mathrm{proj}}_{𝐯}((4,-\mathrm{1,0}))$

The vectors $(\mathrm{1,1},-1)$⁠, $(2,-\mathrm{1,1})$⁠, $(\mathrm{0,1,1})$ form an orthogonal basis. Represent the following vectors in terms of this basis:

a) $(-\mathrm{1,4,0})$

b) $(4,-\mathrm{1,3})$

c) $(-\mathrm{2,6},-2)$

Let $S$ be the subspace spanned by $(2,-\mathrm{1,1})$⁠, $(\mathrm{2,2,2})$⁠. Find an orthogonal basis of $S$ by the Gram–Schmidt process.

Starting with $(\mathrm{1,1},-1)$⁠, $(\mathrm{2,3},-1)$⁠, $(1,-\mathrm{1,1})$⁠, follow the Gram–Schmidt process to find an orthogonal basis of ${ℝ}^3$⁠.

Compute the determinant of the following matrices:

a) $\left[\begin{array}{c}-2\end{array}\right]$

b) $\left[\begin{array}{cc}2& -2\\ 1& 3\end{array}\right]$

c) $\left[\begin{array}{cc}2& 2\\ 2& 2\end{array}\right]$

d) $\left[\begin{array}{ccc}2& 9& -11\\ 0& -1& 5\\ 0& 0& 3\end{array}\right]$

e) $\left[\begin{array}{ccc}2& 1& 0\\ -2& 7& 3\\ 1& 1& 0\end{array}\right]$

f) $\left[\begin{array}{ccc}5& 1& 3\\ 4& 1& 1\\ 4& 5& 1\end{array}\right]$

g) $\left[\begin{array}{cccc}3& 2& 5& 7\\ 0& 0& 2& 0\\ 0& 4& 5& 0\\ 2& 1& 2& 4\end{array}\right]$

h) $\left[\begin{array}{cccc}0& 2& 1& 0\\ 1& 2& -3& 4\\ 5& 6& -7& 8\\ 1& 2& 3& -2\end{array}\right]$

For which $x$ are the following matrices singular (not invertible):

a) $\left[\begin{array}{cc}1& 3\\ 1& x\end{array}\right]$

b) $\left[\begin{array}{cc}3& x\\ 1& 3\end{array}\right]$

c) $\left[\begin{array}{cc}x& 3\\ 3& x\end{array}\right]$

d) $\left[\begin{array}{ccc}x& 1& 0\\ 1& 4& 0\\ 1& 6& 2\end{array}\right]$

Compute

$$\det \left({\left[\begin{array}{cccc}3& 4& 7& 12\\ 0& -1& 9& -8\\ 0& 0& -2& 4\\ 0& 0& 0& 2\end{array}\right]}^{-1}\right)$$

without computing the inverse.

(challenging) Find all the $x$ that make the matrix inverse

$${\left[\begin{array}{cc}1& 2\\ 1& x\end{array}\right]}^{-1}$$

have only integer entries (no fractions). Note that there are two answers.