91Ó°ÊÓ

Determine if the subset of \(C(-\infty, \infty)\) is a subspace of \(C(-\infty, \infty)\) The set of all constant functions: \(f(x)=c\)

Determine whether the set \(S\) is linearly independent or linearly dependent. $$S=\\{(6,2,1),(-1,3,2)\\}$$

Determine whether the set, together with the indicated operations, is a vector space. If it is not, identify at least one of the ten vector space axioms that fails. \(C[0,1]\), the set of all continuous functions defined on the interval \([0,1],\) with the standard operations

Determine if the subset of \(M_{n, n}\) is a subspace of \(M_{n, n}\) with the standard operations. The set of all \(n \times n\) invertible matrices

Find a basis for, and the dimension of, the solution space of \(A \mathbf{x}=\mathbf{0}\) $$A=\left[\begin{array}{rrrr}1 & 4 & 2 & 1 \\ 2 & -1 & 1 & 1 \\ 4 & 2 & 1 & 1 \\ 0 & 4 & 2 & 0\end{array}\right]$$

Let \(V\) be the set of all positive real numbers. Determine whether \(V\) is a vector space with the operations below. $$\begin{aligned}x+y &=x y \\\c x &=x^{c}\end{aligned}$$ If it is, verify each vector space axiom; if not, state all vector space axioms that fail.

Identify and sketch the graph. $$x^{2}+4 y^{2}-16=0$$

Complete the proof of the cancellation property of vector addition by supplying the justification for each step. Prove that if \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are vectors in a vector space \(V\) such that \(\mathbf{u}+\mathbf{w}=\mathbf{v}+\mathbf{w},\) then \(\mathbf{u}=\mathbf{v}\) $$\begin{aligned} \mathbf{u}+\mathbf{w} &=\mathbf{v}+\mathbf{w} & \text { Given } \\\\(\mathbf{u}+\mathbf{w})+(-\mathbf{w}) &=(\mathbf{v}+\mathbf{w})+(-\mathbf{w}) & \text { a. ______ } \\\ \mathbf{u}+(\mathbf{w}+(-\mathbf{w})) &=\mathbf{v}+(\mathbf{w}+(-\mathbf{w})) & \text { b. ______ } \\ \mathbf{u}+\mathbf{0} &=\mathbf{v}+\mathbf{0} & \text { c. ______ } \\ \mathbf{u} &=\mathbf{v} & \text { d. ______ } \end{aligned}$$

Show that the set is linearly dependent by finding a nontrivial linear combination (of vectors in the set) whose sum is the zero vector. Then express one of the vectors in the set as a linear combination of the other vectors in the set. $$S=\\{(1,1,1),(1,1,0),(0,1,1),(0,0,1)\\}$$

Prove that in a given vector space \(V\), the additive inverse of a vector is unique.