Show that u1, u2, u3 form a basis of R3 by proving that they are independent and Span{u1, u2, u3}= R3. Given the following three vectors:\[ \mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \]Show that \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\) form a basis of \(\mathbb{R}^3\) by proving that they are independent and \(\text{Span}\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\} = \mathbb{R}^3\). To justify \(\text{Span}\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\} = \mathbb{R}^3\), show that for a random vector:\[\mathbf{y} = \begin{bmatrix} a \\ b \\ c \end{bmatrix} \in \mathbb{R}^3, \quad \mathbf{y} \in \text{Span}\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}, \, \text{or} \, \begin{bmatrix} a \\ b \\ c \end{bmatrix} = \alpha \mathbf{v}_1 + \beta \mathbf{v}_2 + \gamma \mathbf{v}_3, \, \alpha, \beta, \gamma \in \mathbb{R}\]

Answered: Image /qna-images/answer/559feb33-5439-4f6f-8e73-37c68b935665.jpg

Given the following three vectors: Show that v1, v2, iz form a basis of R by proving that they are independent AND Span {di, ö2, üz} = R*. To justify Span {v1, d2, vz} = R°, show that for a random vector y e R°, ye Span {v1, 02, 03} , or = a•-0j+ß-7z+y•7g, a, ß, y E R

Given the following three vectors: Show that v1, v2, iz form a basis of R by proving that they are independent AND Span {di, ö2, üz} = R*. To justify Span {v1, d2, vz} = R°, show that for a random vector y e R°, ye Span {v1, 02, 03} , or = a•-0j+ß-7z+y•7g, a, ß, y E R

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

See similar textbooks

Similar questions

1) Suppose v1, V2, V3, V4 span a vector space V. Prove that the list v1 - v2, v2 – v3, V3 - V4, V4 also spans V.
Suppose {v1, v2, ..., vp} is an orthogoal set of vectors. Show that {c1v1, c2v2, ..., cpvp} will also be an orthogonal set, for any choice of scalars c1, c2, ..., cp.
7. If Uj + 0 are components of a tensor of the type (0, 2) and if the equation fU¡j +gU ji = 0 holds w.r.t to a basis then prove that eitherf=g and Uj is skew-symmetric orf=-g and Uij is symmetric.
If Span {u,v} = R^2 then {2u,u+v} is a basis for R^2. Is this statement true?
) Consider the vectors -- (-) --- () --- () a = b= = Do these vectors form a basis of R³? If yes, prove that they do, if no explain why.
8. Show that {u,,,} is an orthogonal basis for R³. Then express x as a linear combination of the u's. 2₁ 2 -3 U₂ = 2 3 -=[3³] 5 ₁ and. =-3
suppose that x↓1,..,x↓(n-1) are linearly independent vectors in R^n and also x↓n is notin the span of x↓1,..,x↓(n-1) . 1. prove that x↓1,...,x↓n is a basis of R^n. 2. Prove that if x↓(n+1)∈R^n, then x↓1,..,x↓n,x↓n+1 are not linearly independent
For part b) do we need to check whether the list (x, x, x/2) is also linearly independent? Since a basis of V is a list of vectors in V that is linearly independent and spans V and we know that span = (x, x, x/2) there is also the requirement of linear independence?
Let TilPe > Pz , T (ax?+bx+c) = 2Qx+ b. Then the basis of NIT) is
Q8) Explain why S is not a basis a) S = {(2,1,–2), (-2,–1,2), (4,2, –4)} for R3 b) S = {2,2,x + 3,3x²} for P2
18. Find a basis and dimension of the orthogonal complement to the span ((1, 2, 0, -1), (0, 1, 1, 0), (1, 1, −1, −1)).
Suppose that two vectors z and y in R" are orthogonal, meaning their inner product satisfies z ·g = 0. Show that if both vectors are assumed to be non-zero, then {T, g} are linearly independent. Show that this linear independence fails if at least one of the vectors is the zero vector.