91Ó°ÊÓ

Identify and sketch the graph. $$9 x^{2}+25 y^{2}-36 x-50 y+61=0$$

Determine whether \(S\) is a basis for \(P_{3}\) $$S=\left\\{4-t, t^{3}, 6 t^{2}, t^{3}+3 t, 4 t-1\right\\}$$

Complete the proof of Theorem 4.6 by showing that the intersection of two subspaces of a vector space is closed under scalar multiplication.

Identify and sketch the graph. $$4 x^{2}-y^{2}+4 x+2 y-1=0$$

Prove that if \(A\) is not square, then either the row vectors of \(A\) or the column vectors of \(A\) form a linearly dependent set.

Give an example showing that the rank of the product of two matrices can be less than the rank of either matrix.

Use a graphing utility or computer software program with matrix capabilities to write \(\mathbf{v}\) as a linear combination of \(\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}, \mathbf{u}_{4},\) and \(\mathbf{u}_{5},\) or of \(\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}, \mathbf{u}_{4}, \mathbf{u}_{5},\) and \(\mathbf{u}_{6} .\) Then verify your solution. $$\begin{aligned} \mathbf{u}_{1} &=(1,-3,4,-5,2,-1) \\ \mathbf{u}_{2} &=(3,-2,4,-3,-2,1) \\ \mathbf{u}_{3} &=(1,1,1,-1,4,-1) \\ \mathbf{u}_{4} &=(3,-1,3,-4,2,3) \\ \mathbf{u}_{5} &=(1,-2,1,5,-3,4) \\ \mathbf{u}_{6} &=(4,2,-1,3,-1,1) \\ \mathbf{v} &=(8,17,-16,26,0,-4) \end{aligned}$$

Determine the dimension of the vector space. $$R^{6}$$

Prove each property of the system of linear equations in \(n\) variables \(A \mathbf{x}=\mathbf{b}\) (a) If \(\operatorname{rank}(A)=\operatorname{rank}([A: \mathbf{b}])=n,\) then the system has a unique solution. (b) If \(\operatorname{rank}(A)=\operatorname{rank}([A: \mathbf{b}])

Prove each property of vector addition and scalar multiplication from Theorem 4.2. (a) Property \(1: \mathbf{u}+\mathbf{v}\) is a vector in \(R^{n}\) (b) Property \(2: \mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}\) (c) Property \(3:(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})\) (d) Property \(4: \mathbf{u}+\mathbf{0}=\mathbf{u}\) (e) Property \(5: \mathbf{u}+(-\mathbf{u})=\mathbf{0}\) (f) Property \(6: c \mathbf{u}\) is a vector in \(R^{n}\). (g) Property \(7: c(\mathbf{u}+\mathbf{v})=c \mathbf{u}+c \mathbf{v}\) (h) Property \(8:(c+d) \mathbf{u}=c \mathbf{u}+d \mathbf{u}\) (i) Property \(9: c(d \mathbf{u})=(c d) \mathbf{u}\) (j) Property \(10: 1(\mathbf{u})=\mathbf{u}\)