Linear Algebra Done Right: Solutions and Examples for Success

College
Algebra
Linear algebra

Is ${(1, 4, - 6), (1, 5, 8), (2, 1, 1), (0, 1, 0)}$ a linearly independent subset of $R^{3}$ ?

Marenonigt 2022-01-06

If V is a finite dimensional vector space and W is a subspace, the W is finite dimensional. Prove it.

stop2dance3l 2022-01-06

Label the following statements as being true or false.
(a) There exists a linear operator T with no T-invariant subspace.
(b) If T is a linear operator on a finite-dimensional vector space V, and W is a T-invariant subspace of V, then the characteristic polynomial of Tw divides the characteristic polynomial of T.
(c) Let T be a linear operator on a finite-dimensional vector space V, and let x and y be elements of V. If W is the T-cyclic subspace generated by x, W

Frank Guyton 2022-01-06

Considering that u and v are vectors in W, let W be a subset of the vector space V. If ( $u \oplus v$ ) belongs to W, then W is a subspace of V:
Select one: True or False

prsategazd 2022-01-05

Let V, W, and Z be vector spaces, and let $T : V \to W$ and $U : W \to Z$ be linear.
If UT is onto, prove that U is onto.Must T also be onto?

Marla Payton 2022-01-05

What is the connection between vector functions and space curves?

Carol Valentine 2022-01-05

What is Null Space?

idiopatia0f 2022-01-05

Let V be a vector space, and let $T : V \to V$ be linear. Prove that $T^{2} = T_{0}$ if and only if $R (T) \subseteq N (T) .$

Annette Sabin 2022-01-05

determine whether W is a subspace of the vector space.
$W = {f : f (0) = - 1}, V = C [- 1, 1]$

dedica66em 2022-01-05

Let V and W be vector spaces and $T : V \to W$ be linear. Let ${y_{1}, \dots, y_{k}}$ be a linearly independent subset of $R (T)$ . If $S = {x_{1}, \dots, x_{k}}$ is chosen so that $T (_{ξ}) = y_{i}$ for $i = 1, \dots, k$ , prove that S is linearly independent.

kramtus51 2022-01-05

Find out if the set that can perform the specified operations is a vector space.
Identify the vector space axioms that are false for those that are not vector spaces.
the collection of all real numbers with addition and multiplication operations.
$\circ$ V is not a vector space, and Axioms 7,8,9 fail to hold.
$\circ$ V is not a vector space, and Axiom 6 fails to hold.
$\circ$ V is a vector space.
$\circ$ V is not a vector space, and Axiom 10 fails to hold.
$\circ$ V is not a vector space, and Axioms 6 - 10 fail to hold.

Linda Seales 2022-01-05

Show that the xy plane $W = (x, y, 0)$ in $R^{3}$ is generated by
(i) $u = [\begin{matrix} 1 \\ 2 \\ 0 \end{matrix}]$ and $v = [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}]$ (ii) $u = [\begin{matrix} 2 \\ - 1 \\ 0 \end{matrix}]$ and $v = [\begin{matrix} 1 \\ 3 \\ 0 \end{matrix}]$

Kathy Williams 2022-01-05

Find the dimension of the vector space U of all linear transformations of V into W for each of the following:
(a) $V = R^{2}, W = R^{3}$
(b) $V = P_{2}, W = P_{1}$
(c) $V = M_{21}, W = M_{32}$
(d) $V = R_{3}, W = R_{4}$

Find the directional derivative of at a given point in the direction indicated by the angle theta. F (x,y)=x^3y^4+x^4y^3, (1,1)

Nicontio1 2022-01-04

Let V, W, and Z be vector spaces, and let $T : V \to W$ and $U : W \to Z$ be linear.
If UT is one-to-one, prove that T is one-to-one. must U also be one-to-one?

Holly Guerrero 2022-01-04

Problem 2:
If $V = R^{3}$ is a vector space and let H be a subset of V and is defined as $H = {(a, b, c) : c^{2} + b^{2} = 0, a \geq 0}$ . Show that H is not subspace of vector space
Problem 3
Let $V = R^{3}$ be a vector space and let W be a subset of V, where $W = {(a, b, c) : a^{2} = b^{2}}$ . Determine whether W is a subspace of vector space or not.

widdonod1t 2022-01-04

(a) Let U and W be subspaces of a vector space V. Prove that $U \cap W = {v : v \in U and v \in W}$ is a subspace of V.
(b) Give an example of two subspaces U and W and a vector space V such that $U \cup W = {v : v \in U or v \in W}$ is not a subspace of V.

killjoy1990xb9 2022-01-04

Prove that
If W is a subspace of a vector space V and $w_{1}, w_{2}, \dots, w_{n}$ are in W, then $a_{1} w_{1} + a_{2} w_{2} + \dots + a_{n} w_{n} \in W$ for any scalars $a_{1}, a_{2}, \dots, a_{n}$ .

Betsy Rhone 2022-01-04

Determine whether the set equipped with the given operations is a vector space.
For those that are not vector spaces identify the vector space axioms that fail.
The set of all pairs of real numbers of the form (x,0) with the standard operations on $R^{2}$ .
$\circ$ V is a vector space.
$\circ$ V is not a vector space, and Axiom 7,8, 9 fails to hold.
$\circ$ V is not a vector space, and Axioms 4 and 5 fail to hold.
$\circ$ V is not a vector space, and Axioms 2 and 3 fail to hold.
$\circ$ V is not a vector space, and Axiom 10 fails to hold.

interdicoxd 2022-01-04

For which value of $k$ the vector $u = [\begin{matrix} 1 \\ - 2 \\ k \end{matrix}]$ is a linear combination of the vectors $v_{1} = [\begin{matrix} 3 \\ 0 \\ - 2 \end{matrix}]$ and $v_{2} = [\begin{matrix} 2 \\ - 1 \\ - 5 \end{matrix}]$ .

Finding detailed linear algebra problems and solutions has always been difficult because the textbooks would never provide anything that would be sufficient. Since it is used not only by engineering students but by anyone who has to work with specific calculations, we have provided you with a plethora of questions and answers in their original form. It will help you to see some logic as you are solving complex numbers and understand the basic concepts of linear Algebra in a clearer way. If you need additional help or would like to connect several solutions, compare more than one solution as you approach your task.

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