# Stanford-university-exam-eye-catch

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- Linear Independent Vectors, Invertible Matrix, and Expression of a Vector as a Linear Combinations Consider the matrix \[A=\begin{bmatrix} 1 & 2 & 1 \\ 2 &5 &4 \\ 1 & 1 & 0 \end{bmatrix}.\] (a) Calculate the inverse matrix $A^{-1}$. If you think the matrix $A$ is not invertible, then explain why. (b) Are the vectors \[ […]
- If Every Vector is Eigenvector, then Matrix is a Multiple of Identity Matrix Let $A$ be an $n\times n$ matrix. Assume that every vector $\mathbf{x}$ in $\R^n$ is an eigenvector for some eigenvalue of $A$. Prove that there exists $\lambda\in \R$ such that $A=\lambda I$, where $I$ is the $n\times n$ identity matrix. Proof. Let us write […]
- Find the Nullity of the Matrix $A+I$ if Eigenvalues are $1, 2, 3, 4, 5$ Let $A$ be an $n\times n$ matrix. Its only eigenvalues are $1, 2, 3, 4, 5$, possibly with multiplicities. What is the nullity of the matrix $A+I_n$, where $I_n$ is the $n\times n$ identity matrix? (The Ohio State University, Linear Algebra Final Exam […]
- Inequality about Eigenvalue of a Real Symmetric Matrix Let $A$ be an $n\times n$ real symmetric matrix. Prove that there exists an eigenvalue $\lambda$ of $A$ such that for any vector $\mathbf{v}\in \R^n$, we have the inequality \[\mathbf{v}\cdot A\mathbf{v} \leq \lambda \|\mathbf{v}\|^2.\] Proof. Recall […]
- A Linear Transformation is Injective (One-To-One) if and only if the Nullity is Zero Let $U$ and $V$ be vector spaces over a scalar field $\F$. Let $T: U \to V$ be a linear transformation. Prove that $T$ is injective (one-to-one) if and only if the nullity of $T$ is zero. Definition (Injective, One-to-One Linear Transformation). A linear […]
- If matrix product $AB$ is a square, then is $BA$ a square matrix? Let $A$ and $B$ are matrices such that the matrix product $AB$ is defined and $AB$ is a square matrix. Is it true that the matrix product $BA$ is also defined and $BA$ is a square matrix? If it is true, then prove it. If not, find a […]
- Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$ Determine whether there exists a nonsingular matrix $A$ if \[A^4=ABA^2+2A^3,\] where $B$ is the following matrix. \[B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 2 & 1 & -4 \end{bmatrix}.\] If such a nonsingular matrix $A$ exists, find the inverse […]
- Show the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis Let $P_3$ be the vector space over $\R$ of all degree three or less polynomial with real number coefficient. Let $W$ be the following subset of $P_3$. \[W=\{p(x) \in P_3 \mid p'(-1)=0 \text{ and } p^{\prime\prime}(1)=0\}.\] Here $p'(x)$ is the first derivative of $p(x)$ and […]