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Linear Algebra exam problems and solutions at University of California, Berkeley


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  • Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose)Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose) Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.   Hint. Recall that the rank of a matrix $A$ is the dimension of the range of $A$. The range of $A$ is spanned by the column vectors of the matrix […]
  • Union of Subspaces is a Subspace if and only if One is Included in AnotherUnion of Subspaces is a Subspace if and only if One is Included in Another Let $W_1, W_2$ be subspaces of a vector space $V$. Then prove that $W_1 \cup W_2$ is a subspace of $V$ if and only if $W_1 \subset W_2$ or $W_2 \subset W_1$.     Proof. If $W_1 \cup W_2$ is a subspace, then $W_1 \subset W_2$ or $W_2 \subset […]
  • Diagonalize the 3 by 3 Matrix if it is DiagonalizableDiagonalize the 3 by 3 Matrix if it is Diagonalizable Determine whether the matrix \[A=\begin{bmatrix} 0 & 1 & 0 \\ -1 &0 &0 \\ 0 & 0 & 2 \end{bmatrix}\] is diagonalizable. If it is diagonalizable, then find the invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.   How to […]
  • Is the Following Function $T:\R^2 \to \R^3$ a Linear Transformation?Is the Following Function $T:\R^2 \to \R^3$ a Linear Transformation? Determine whether the function $T:\R^2 \to \R^3$ defined by \[T\left(\, \begin{bmatrix} x \\ y \end{bmatrix} \,\right) = \begin{bmatrix} x_+y \\ x+1 \\ 3y \end{bmatrix}\] is a linear transformation.   Solution. The […]
  • Matrices Satisfying $HF-FH=-2F$Matrices Satisfying $HF-FH=-2F$ Let $F$ and $H$ be an $n\times n$ matrices satisfying the relation \[HF-FH=-2F.\] (a) Find the trace of the matrix $F$. (b) Let $\lambda$ be an eigenvalue of $H$ and let $\mathbf{v}$ be an eigenvector corresponding to $\lambda$. Show that there exists an positive integer $N$ […]
  • Find All 3 by 3 Reduced Row Echelon Form Matrices of Rank 1 and 2Find All 3 by 3 Reduced Row Echelon Form Matrices of Rank 1 and 2 (a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1. (b) Find all such matrices with rank 2.   Solution. (a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1. First we look at the rank 1 case. […]
  • Basis of Span in Vector Space of Polynomials of Degree 2 or LessBasis of Span in Vector Space of Polynomials of Degree 2 or Less Let $P_2$ be the vector space of all polynomials of degree $2$ or less with real coefficients. Let \[S=\{1+x+2x^2, \quad x+2x^2, \quad -1, \quad x^2\}\] be the set of four vectors in $P_2$. Then find a basis of the subspace $\Span(S)$ among the vectors in $S$. (Linear […]
  • A Maximal Ideal in the Ring of Continuous Functions and a Quotient RingA Maximal Ideal in the Ring of Continuous Functions and a Quotient Ring Let $R$ be the ring of all continuous functions on the interval $[0, 2]$. Let $I$ be the subset of $R$ defined by \[I:=\{ f(x) \in R \mid f(1)=0\}.\] Then prove that $I$ is an ideal of the ring $R$. Moreover, show that $I$ is maximal and determine […]

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