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# Linear algebra: test 1

Test 1 for Linear algebra: course

7 problems, 10 points each

Instructions: Show enough work to justify your answers. Refer to theorems in the book whenever necessary.

1. Prove that the set of all non-zero rational numbers, $\mathbf{Q}\backslash \{0\},$ is closed under division.
2. Prove that the intersection of two subspaces is always a subspace.
3. Prove that the set of all diagonal $n\times n$ matrices, i.e., ones with $a_{ij}=0$ for all $i\neq j,$ is a subspace of $M(n,n).$
4. Find the reduced row echelon form of the following system of linear equations. What is the dimension of its solution set? $$x+y-z=1$$ $$x-y+2z=0$$ $$3x+y\;\;\;\;\;=2$$
5. Is it possible that a homogeneous system of linear equations has (a) no solutions, (b) one solution, (c) two solutions, (3) infinitely many solutions? Give an example of such a system or explain why it's not possible.
6. The set $$\{(2,1,1),(1,2,1),(0,1,1),(1,1,1)\}$$ spans $\mathbf{R}^{3}.$ Use the proof of the Reduction Theorem as a recipe for finding a subset of this set that is a basis of $\mathbf{R}^{3}.$
7. Let $$B=\{x-1,x+1\}.$$ (a) Prove that $B$ is a basis of $\mathbf{P}_{1}.$ (b) Find the coordinate vector $[3x-5]_{B.}$