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 […]
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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 […]
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 […]
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 […]
