91Ó°ÊÓ

Determine whether the vectors emanating from the origin and terminating at the following pairs of points are parallel.(a) \((3,1,2)\) and \((6,4,2)\) (b) \((-3,1,7)\) and \((9,-3,-21)\) (c) \((5,-6,7)\) and \((-5,6,-7)\) (d) \((2,0,-5)\) and \((5,0,-2)\)

Label the following statements as true or false. (a) If \(V\) is a vector space and \(W\) is a subset of \(V\) that is a vector space, then \(W\) is a subspace of \(V\). (b) The empty set is a subspace of every vector space. (c) If \(V\) is a vector space other than the zero vector space, then \(V\) contains a subspace \(W\) such that \(W \neq V\). (d) The intersection of any two subsets of \(V\) is a subspace of \(V\). (e) An \(n \times n\) diagonal matrix can never have more than \(n\) nonzero entries. (f) The trace of a square matrix is the product of its diagonal entries. (g) Let \(\mathrm{W}\) be the \(x y\)-plane in \(\mathrm{R}^{3}\); that is, \(\mathrm{W}=\left\\{\left(a_{1}, a_{2}, 0\right): a_{1}, a_{2} \in R\right\\}\). Then \(W=R^{2}\).

Determine the transpose of each of the matrices that follow. In addition, if the matrix is square, compute its trace. (a) \(\left(\begin{array}{rr}-4 & 2 \\ 5 & -1\end{array}\right)\) (b) \(\left(\begin{array}{rrr}0 & 8 & -6 \\ 3 & 4 & 7\end{array}\right)\) (c) \(\left(\begin{array}{rr}-3 & 9 \\ 0 & -2 \\ 6 & 1\end{array}\right)\) (d) \(\left(\begin{array}{rrr}10 & 0 & -8 \\ 2 & -4 & 3 \\ -5 & 7 & 6\end{array}\right)\) (e) \(\left(\begin{array}{llll}1 & -1 & 3 & 5\end{array}\right)\) (f) \(\left(\begin{array}{rrrr}-2 & 5 & 1 & 4 \\ 7 & 0 & 1 & -6\end{array}\right)\) (g) \(\left(\begin{array}{l}5 \\ 6 \\ 7\end{array}\right)\) (h) \(\left(\begin{array}{rrr}-4 & 0 & 6 \\ 0 & 1 & -3 \\ 6 & -3 & 5\end{array}\right)\)

Show that the set of convergent sequences is an infinite-dimensional subspace of the vector space of all sequences of real numbers. (See Exercise 21 in Section 1.3.)

Write the zero vector of \(M_{3 \times 4}(F)\).

For each of the following lists of vectors in \(\mathrm{R}^{3}\), determine whether the first vector can be expressed as a linear combination of the other two. (a) \((-2,0,3),(1,3,0),(2,4,-1)\) (b) \((1,2,-3),(-3,2,1),(2,-1,-1)\) (c) \((3,4,1),(1,-2,1),(-2,-1,1)\) (d) \((2,-1,0),(1,2,-3),(1,-3,2)\) (e) \((5,1,-5),(1,-2,-3),(-2,3,-4)\) (f) \((-2,2,2),(1,2,-1),(-3,-3,3)\)

Let \(V\) be the set of real numbers regarded as a vector space over the field of rational numbers. Prove that \(\mathrm{V}\) is infinite-dimensional. Hint: Use the fact that \(\pi\) is transcendental, that is, \(\pi\) is not a zero of any polynomial with rational coefficients.

Determine which of the following sets are bases for \(\mathrm{P}_{2}(R)\). (a) \(\left\\{-1-x+2 x^{2}, 2+x-2 x^{2}, 1-2 x+4 x^{2}\right\\}\) (b) \(\left\\{1+2 x+x^{2}, 3+x^{2}, x+x^{2}\right\\}\) (c) \(\left\\{1-2 x-2 x^{2},-2+3 x-x^{2}, 1-x+6 x^{2}\right\\}\) (d) \(\left\\{-1+2 x+4 x^{2}, 3-4 x-10 x^{2},-2-5 x-6 x^{2}\right\\}\) (e) \(\left\\{1+2 x-x^{2}, 4-2 x+x^{2},-1+18 x-9 x^{2}\right\\}\)

Let \(W\) be a subspace of a (not necessarily finite-dimensional) vector space \(V\). Prove that any basis for \(W\) is a subset of a basis for \(V\).

Prove that if the vector \(x\) emanates from the origin of the Euclidean plane and terminates at the point with coordinates \(\left(a_{1}, a_{2}\right)\), then the vector \(t x\) that emanates from the origin terminates at the point with coordinates \(\left(t a_{1}, t a_{2}\right)\). Visit goo.gl/eYTxuU for a solution.