pdf: [[Gilbert Strang - Introduction to Linear Algebra Fifth Edition-WELLESLEY -CAMBRIDGE PRESS (2016 ).pdf]] course: [[Linear Algebra]] ## Problems ### 1 Introduction to Vectors #### 1.1 Vectors and Linear Combinations 1. Describe geometrically (line, plane, or all of $\mathbb{R}^{3}$ ) all linear combinations of (a) $\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]$ and $\left[\begin{array}{l}3 \\ 6 \\ 9\end{array}\right] \quad$ (b) $\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ and $\left[\begin{array}{l}0 \\ 2 \\ 3\end{array}\right] \quad$ (c) $\left[\begin{array}{l}2 \\ 0 \\ 0\end{array}\right]$ and $\left[\begin{array}{l}0 \\ 2 \\ 2\end{array}\right]$ and $\left[\begin{array}{l}2 \\ 2 \\ 3\end{array}\right]$ 2. Draw $\boldsymbol{v}=\left[\begin{array}{l}4 \\ 1\end{array}\right]$ and $\boldsymbol{w}=\left[\begin{array}{r}-2 \\ 2\end{array}\right]$ and $v+w$ and $v-w$ in a single $x y$ plane. 3. If $\boldsymbol{v}+\boldsymbol{w}=\left[\begin{array}{l}5 \\ 1\end{array}\right]$ and $\boldsymbol{v}-\boldsymbol{w}=\left[\begin{array}{l}1 \\ 5\end{array}\right]$, compute and draw the vectors $\boldsymbol{v}$ and $\boldsymbol{w}$ 4. From $v=\left[\begin{array}{l}2 \\ 1\end{array}\right]$ and $w=\left[\begin{array}{l}1 \\ 2\end{array}\right]$, find the components of $3 v+w$ and $c v+d w$ 5. Compute $u+v+w$ and $2 u+2 v+w .$ How do you know $u, v, w$ lie in a plane? $\boldsymbol{u}=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right], \quad \boldsymbol{v}=\left[\begin{array}{r}-3 \\ 1 \\ -2\end{array}\right], \quad \boldsymbol{w}=\left[\begin{array}{r}2 \\ -3 \\ -1\end{array}\right]$ 6. Every combination of $v=(1,-2,1)$ and $w=(0,1,-1)$ has components that add to $\underline{\qquad\qquad}$. Find $c$ and $d$ so that $c v+d w=(3,3,-6) .$ Why is $(3,3,6)$ impossible? 7. In the $x y$ plane mark all nine of these linear combinations: $c \left[\begin{array}{l}2 \\ 1\end{array}\right]+d\left[\begin{array}{l}0 \\ 1\end{array}\right]$ with $c=0,1,2$ and $d=0,1,2 .$ 8. The parallelogram with sides $v$ and $w$ has diagonal $v+w$. What is its other diagonal? What is the sum of the two diagonals? Draw that vector sum. 9. If three corners of a parallelogram are $(1,1),(4,2)$, and $(1,3)$, what are all three of the possible fourth corners? Draw two of them. 10. Which point of the cube is $i+j$ ? Which point is the vector sum of $i=(1,0,0)$ and $j=(0,1,0)$ and $\boldsymbol{k}=(0,0,1)$ ? Describe all points $(x, y, z)$ in the cube. #needs-diagram 11. Four corners of this unit cube are $(0,0,0),(1,0,0),(0,1,0),(0,0,1)$. What are the other four corners? Find the coordinates of the center point of the cube. The center points of the six faces are $\underline{\qquad\qquad}.$ The cube has how many edges? #needs-diagram 12. In $x y z$ space, where is the plane of all linear combinations of $i=(1,0,0)$ and $i+j=(1,1,0)$ ? #needs-diagram ![[Screen Shot 2022-04-20 at 2.34.31 PM.png]] 13. (3 Parts) #needs-diagram - What is the sum $V$ of the twelve vectors that go from the center of a clock to the hours $1: 00,2: 00, \ldots, 12: 00$ ? - If the $2: 00$ vector is removed, why do the 11 remaining vectors add to $8: 00$ ? - What are the $x, y$ components of that $2: 00$ vector $v=(\cos \theta, \sin \theta) ?$ 14. Suppose the twelve vectors start from $6: 00$ at the bottom instead of $(0,0)$ at the center. The vector to $12: 00$ is doubled to $(0,2)$. The new twelve vectors add to $\underline{\qquad\qquad}.$ #needs-diagram 15. Figure $1.5$ a shows $\frac{1}{2} v+\frac{1}{2} \boldsymbol{w}$. Mark the points $\frac{3}{4} \boldsymbol{v}+\frac{1}{4} \boldsymbol{w}$ and $\frac{1}{4} \boldsymbol{v}+\frac{1}{4} \boldsymbol{w}$ and $\boldsymbol{v}+\boldsymbol{w}$. 16. Mark the point $-\boldsymbol{v}+2 \boldsymbol{w}$ and any other combination $c v+d \boldsymbol{w}$ with $c+d=1$. Draw the line of all combinations that have $c+d=1$ 17. Locate $\frac{1}{3} \boldsymbol{v}+\frac{1}{3} \boldsymbol{w}$ and $\frac{2}{3} \boldsymbol{v}+\frac{2}{3} \boldsymbol{w}$. The combinations $c \boldsymbol{v}+c \boldsymbol{w}$ fill out what line? 18. Restricted by $0 \leq c \leq 1$ and $0 \leq d \leq 1$, shade in all combinations $c v+d \boldsymbol{w}$. 19. Restricted only by $c \geq 0$ and $d \geq 0$ draw the "cone" of all combinations $c v+d \boldsymbol{w}$ #needs-diagram ![[Screen Shot 2022-04-20 at 2.33.34 PM.png]] 20. Locate $\frac{1}{3} \boldsymbol{u}+\frac{1}{3} \boldsymbol{v}+\frac{1}{3} \boldsymbol{w}$ and $\frac{1}{2} \boldsymbol{u}+\frac{1}{2} \boldsymbol{w}$ in Figure $1.5 \mathrm{~b} .$ - Challenge problem: Under what restrictions on $c, d, e$, will the combinations $c u+d v+e \boldsymbol{w}$ fill in the dashed triangle? To stay in the triangle, one requirement is $c \geq 0, d \geq 0, e \geq 0$ 21. The three sides of the dashed triangle are $v-u$ and $\boldsymbol{w}-\boldsymbol{v}$ and $\boldsymbol{u}-\boldsymbol{w}$. Their sum is $\underline{\qquad\qquad}$ plus $(-2,-2)$ Draw the head-to-tail addition around a plane triangle of $(3,1)$ plus $(-1,1)$ 22. Shade in the pyramid of combinations $c u+d v+e w$ with $c \geq 0, d \geq 0, e \geq 0$ and $c+d+e \leq 1 .$ Mark the vector $\frac{1}{2}(u+v+w)$ as inside or outside this pyramid. 23. If you look at all combinations of those $u, v$, and $w$, is there any vector that can't be produced from $c u+d v+e w$ ? Different answer if $u, v, w$ are all in $\underline{\qquad\qquad}$ 24. Which vectors are combinations of $\boldsymbol u$ and $\boldsymbol v$, and $a l s o$ combinations of $v$ and $\boldsymbol{w}$? 25. Draw vectors $\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}$ so that their combinations $c \boldsymbol{u}+d \boldsymbol{v}+e \boldsymbol{w}$ fill only a line. Find vectors $\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}$ so that their combinations $c \boldsymbol{u}+d \boldsymbol{v}+e \boldsymbol{w}$ fill only a plane. 26. What combination $c\left[\begin{array}{c}1 \\ 2\end{array}\right]+d\left[\begin{array}{c}3 \\ 1\end{array}\right]$ produces $\left[\begin{array}{c}14 \\ 8\end{array}\right]$ ? Express this question as two equations for the coefficients $c$ and $d$ in the linear combination. 27. How many corners does a cube have in 4 dimensions? How many 3D faces? How many edges? A typical corner is $(0,0,1,0)$. A typical edge goes to $(0,1,0,0)$. 28. Find vectors $v$ and $w$ so that $v+w=(4,5,6)$ and $v-w=(2,5,8) .$ This is a question with $\underline{\qquad\quad}$ unknown numbers, and an equal number of equations to find those numbers. 29. Find two different combinations of the three vectors $\boldsymbol{u}=(1,3)$ and $\boldsymbol{v}=(2,7)$ and $\boldsymbol{w}=(1,5)$ that produce $\boldsymbol{b}=(0,1)$. Slightly delicate question: If I take any three vectors $\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}$ in the plane, will there always be two different combinations that produce $\boldsymbol{b}=(0,1)$ ? 30. The linear combinations of $\boldsymbol{v}=(a, b)$ and $\boldsymbol{w}=(c, d)$ fill the plane unless Find four vectors $\boldsymbol{u}, \boldsymbol{v}, \boldsymbol{w}, \boldsymbol{z}$ with four components each so that their combinations $c \boldsymbol{u}+d \boldsymbol{v}+e \boldsymbol{w}+f \boldsymbol{z}$ produce all vectors $\left(b_{1}, b_{2}, b_{3}, b_{4}\right)$ in four-dimensional space. 31. Write down three equations for $c, d, e$ so that $c\boldsymbol u+d\boldsymbol v+e\boldsymbol w=b$. Can you somehow find $c, d, e$ for this $b$ ? $\boldsymbol u=\left[\begin{array}{r}2 \\ -1 \\ 0\end{array}\right] \quad \boldsymbol v=\left[\begin{array}{r}-1 \\ 2 \\ -1\end{array}\right] \quad \boldsymbol w=\left[\begin{array}{r}0 \\ -1 \\ 2\end{array}\right] \quad \boldsymbol b=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ #### 1.2 Lengths and Dot Products I will not be able to do this alone. Need intern #### 1.3 Matrices ### 2 Solving Linear Equations #### 2.1 Vectors and Linear Equations #### 2.2 The Idea of Elimination #### 2.3 Elimination Using Matrices #### 2.4 Rules for Matrix Operations #### 2.5 Inverse Matrices #### 2.6 Elimination = Factorization: A = LU #### 2.7 Transposes and Permutations ### 3 Vector Spaces and Subspaces #### 3.1 Spaces of Vectors #### 3.2 The Nullspace of A: Solving Ax = 0 and Ra = 0 #### 3.3 The Complete Solution to Ax = b #### 3.4 Independence, Basis and Dimension #### 3.5 Dimensions of the Four Subspaces ### 4 Orthogonality #### 4.1 Orthogonality of the Four Subspaces #### 4.2 Projections #### 4.3 Least Squares Approximations #### 4.4 Orthonormal Bases and Gram-Schmidt ### 5 Determinants #### 5.1 The Properties of Determinants #### 5.2 Permutations and Cofactors #### 5.3 Cramer's Rule, Inverses, and Volumes ### 6 Eigenvalues and Eigenvectors #### 6.1 Introduction to Eigenvalues #### 6.2 Diagonalizing a Matrix #### 6.3 Systems of Differential Equations #### 6.4 Symmetric Matrices #### 6.5 Positive Definite Matrices ### 7 The Singular Value Decomposition (SVD) #### 7.1 Image Processing by Linear Algebra #### 7.2 Bases and Matrices in the SVD #### 7.3 Principal Component Analysis (PCA by the SVD) #### 7.4 The Geometry of the SVD ### 8 Linear Transformations #### 8.1 The Idea of a Linear Transformation #### 8.2 The Matrix of a Linear Transformation #### 8.3 The Search for a Good Basis ### 9 Complex Vectors and Matrices #### 9.1 Complex Numbers #### 9.2 Hermitian and Unitary Matrices #### 9.3 The Fast Fourier Transform ### 10 Applications #### 10.1 Graphs and Networks #### 10.2 Matrices in Engineering #### 10.3 Markov Matrices, Population, and Economics #### 10.4 Linear Programming #### 10.5 Fourier Series: Linear Algebra for Functions #### 10.6 Computer Graphics #### 10.7 Linear Algebra for Cryptography ### 11 Numerical Linear Algebra #### 11.1 Gaussian Elimination in Practice #### 11.2 Norms and Condition Numbers #### 11.3 Iterative Methods and Preconditioners ### 12 Linear Algebra in Probability & Statistics #### 12.1 Mean, Variance, and Probability #### 12.2 Covariance Matrices and Joint Probabilities #### 12.3 Multivariate Gaussian and Weighted Least Squares ### Matrix Factorizations