>[!note]- Raw Exam Without Helpful Context(Realistic Test Environment) >![[Realistic, AS IS Exam without Context]] >[!tip]- Lectures On The Problems With Answers >![[Lecture Exam 2]] --- # Linear Algebra - Practice Sheet **Note:** Throughout this assignment, the notations $\vec{x}$ and $\mathbf{x}$ are equivalent and represent vectors. ## Section 1: Inner Products and Orthogonality (Vector Relationships) ### Problem 1: Vector Orthogonality (Perpendicular Vectors in $\mathbb{R}^3$) Let $\mathrm{x}=(1,-1,2), \mathrm{y}=(1,3,1)$. Show that $\vec{x}$ and $\vec{y}$ are orthogonal in $\mathbb{R}^3$ with respect to the dot product (i.e., scalar product). <div style="page-break-after: always;"></div> ![[Notebook - Exam 2 Targeted Studies 03-23-2025-01.svg]] %%[[Notebook - Exam 2 Targeted Studies 03-23-2025-01.md|🖋 Edit in Excalidraw]]%% ### Problem 2: Vector Distance (Separation Between Points in $\mathbb{R}^2$) Let $\mathbf{x}=(2,-2), \mathbf{y}=(1,-3)$. Compute the distance between $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb{R}^2$ with respect to the dot product (i.e., scalar product). <div style="page-break-after: always;"></div> ## Section 2: Function Spaces ($L^2$ Inner Products) ### Problem 3: Function Orthogonality (Perpendicular Functions in $C[0,1]$) Let $f(x)=x-3/4, g(x)=x^2$. Show that $f$ and $g$ are orthogonal in $C[0,1]$ with respect to the $L^2(dx)$ inner product. <div style="page-break-after: always;"></div> ### Problem 4: Norm Calculation (Function Length Measurement) Let $f(x)=e^x$. Compute $|f|$ in $C[0,2]$ with respect to the $L^2(dx)$ inner product. <div style="page-break-after: always;"></div> ## Section 3: Optimization (Minimization Problems) ### Problem 5: Closest Point (Minimizing Distance to a Line) Find the point on the line $y=3x$ that is closest to the point $(1,1)$ with respect to the dot product (i.e., scalar product) on $\mathbb{R}^2$. <div style="page-break-after: always;"></div> ## Section 4: Least Squares (Data Fitting) ### Problem 6: Normal Equations (Best Quadratic Approximation) Set up and simplify the normal equations (you do not need to solve them) for best quadratic least squares fit to the data: $\begin{array}{c|c|c|c|c} \mathrm{x} & -1 & 0 & 1 & 2 \\ \hline \mathrm{y} & 4 & 1 & 0 & 0 \end{array} $ <div style="page-break-after: always;"></div> ## Section 5: Subspaces and Linear Transformations (Vector Space Structures) ### Problem 7: Subspace Verification (Testing Closure Properties) Show that the set of functions $f$ in $C^1[0,1]$ such that $f^{\prime}(0)=f^{\prime}(1)$ form a subspace of $C^1[0,1]$ (recall that $C^1[0,1]$ is the set of continuous functions with continuous first derivative on $[0,1]$). <div style="page-break-after: always;"></div> ### Problem 8: Linear Transformation (Function Mapping Properties) For each $f \in C[0,1]$ define the transformation $L(f(x))=x^3 f(x)$. Show that $L$ defines a linear transformation on $C[0,1]$. <div style="page-break-after: always;"></div> ## Section 6: Matrix Spaces (Foundational Vector Spaces) ### Problem 9: Fundamental Spaces (Key Subspaces of a Matrix) Consider the matrix $A$ and its reduced row echelon form, $\operatorname{rref}(A)$, as given by: $ A=\left(\begin{array}{ccc} 2 & 2 & 1 \\ -4 & -2 & -3 \\ 5 & 4 & 3 \end{array}\right), \quad \operatorname{rref}(A)=\left(\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & \frac{-1}{2} \\ 0 & 0 & 0 \end{array}\right) $ #### Part (a) Find a basis for the row space of $A$ (set of all possible linear combinations of rows). <div style="page-break-after: always;"></div> #### Part (b) Find a basis for the column space of $A$ (range of the linear transformation). <div style="page-break-after: always;"></div> #### Part (c) Find a basis for the null space of $A$ (solution space to $A\mathbf{x}=\mathbf{0}$). <div style="page-break-after: always;"></div> ## Section 7: Linear Independence and Span (Vector Relationships) ### Problem 10: Vector Space Properties (Analyzing Vector Sets) Let $\mathbf{v}_1=(1,-1,2), \mathbf{v}_2=(1,1,1), \mathbf{v}_3=(1,-3,3), \mathbf{b}=(-1,7,-5)$. #### Part (a) Is $\mathbf{b}$ in the span of $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$? (Can $\mathbf{b}$ be written as a linear combination of these vectors?) Show your work and explain how your work justifies your answer. <div style="page-break-after: always;"></div> #### Part (b) Are $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ linearly independent or dependent? (Is there a non-trivial way to combine them to get zero?) Explain how your work justified your answer. <div style="page-break-after: always;"></div> #### Part (c) What is the dimension of the span of $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$? (How many vectors are needed in a basis?) Explain how your work justifies your answer. <div style="page-break-after: always;"></div>