91Ó°ÊÓ

What \(2 \times 2\) matrices \(A\) satisfy $$ A^{2}=0, \quad A^{2}=I, \quad A^{2}=-I ? $$

defined and well defined? (a) "The aunt of," from people to people. (b) \(f(x)=\frac{1}{x}\), from real numbers to real numbers. (c) "The capital of, \(^{\text {" }}\) from countries to cities (careful-at least two countries, the Netherlands and Bolivia, have two capitals.)

defined and well defined? (a) "The aunt of," from people to people. (b) \(f(x)=\frac{1}{x}\), from real numbers to real numbers. (c) "The capital of, \(^{\text {" }}\) from countries to cities (careful-at least two countries, the Netherlands and Bolivia, have two capitals.)

dimensions of the corresponding matrix? (a) \(T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}\) (b) \(T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) (c) \(T: \mathbb{R}^{4} \rightarrow \mathbb{R}^{2}\) (d) \(T: \mathbb{R}^{4} \rightarrow \mathbb{R}\)

assignments, and the final were entered in matrix form, each row corresponding to a student, the first column corresponding to the grade on the mid-term, the next 10 columns corresponding to grades on the homeworks and the last column corresponding to the grade on the final. The final counts for 50 percent, the mid-term counts for 25 percent, and each homework for \(1.5\) percent of the final grade. What is the transformation \(T: \mathbb{R}^{12} \rightarrow \mathbb{R}\) that assigns to each student his or her final grade?

of \(x\). (a) \(f(x)=x^{2}-1, g(x)=3 x, h(x)=-x+2\), for \(x=2\).(b) \(f(x)=x^{2}, g(x)=x-3, h(x)=x-3\), for \(x=1\)

If \(A\) and \(B\) are \(n \times n\) matrices, their Jordan product is $$ \frac{A B+B A}{2} $$ this product is commutative but not associative.

\(w \in \mathbb{C}\) is a subspace of \(\mathbb{R}^{2}=\mathbb{C}\). Describe this subspace.

If \(\overrightarrow{\mathbf{v}}\) and \(\vec{w}\) are vectors, and \(A\) is a matrix, which of the following are rs? Which are vectors? $$ \overrightarrow{\mathbf{v}} \times \overrightarrow{\mathbf{w}} ; \quad \overrightarrow{\mathbf{v}} \cdot \overrightarrow{\mathbf{w}} ; \quad|\overrightarrow{\mathbf{v}}| ; \quad|A| ; \operatorname{det} A ; \quad A \overrightarrow{\mathbf{v}} $$

If \(\vec{v}\) and \(\vec{w}\) are vectors, and \(A\) is a matrix, which of the following are 8? Which are vectors? $$ \overrightarrow{\mathbf{v}} \times \vec{w} ; \quad \vec{v} \cdot \vec{w} ; \quad|\vec{v}| ; \quad|A| ; \quad \operatorname{det} A_{i} \quad A \vec{v} $$