a) Determine whether or not the set of vectors is a basis of the indicated space: x²-1, x*+x+1,x),P; b) Find a basis, and the dimension, of the vector space of all matrices of order 3. c) Find the coordinates of the vector (2, -3, 4, -5) relative to the basis (0, 1, 0, -1), (1, 0, -1, 1), (0, 0, -1, 0), (-1, 1, -1, 0) of R'.

Q: If {u, v} is a basis for a vector space V, show that {u – 2v, au} is also a basis for V for every…

A: Here we use the definition of basis and linear combination of vectors as follows

Q: The vectors u1 = (1, 1, 1, 1), u2 = (0, 1, 1, 1), u3 = (0, 0, 1, 1), and u4 = (0, 0, 0, 1) form a…

A: Given,u1 = (1, 1, 1, 1), u2 = (0, 1, 1, 1), u3 = (0, 0, 1, 1), and u4 = (0, 0, 0, 1)Consider the…

Q: Let V = Mmxn(C) be the set of m x n matrices with entries in C. 1. Describe a basis for V as a…

A: For the solution of the problem follow the next steps.

Q: Find a basis for the space spanned by the given vectors. 2 3 2 -2 3 -2 -1 -1 b. 6 -9 -4 -14 а. -2 -8…

A: See the detailed solution below.

Q: Does the set of vectors {(1,3, -2), (-3, - 12, 10), (-2, -6, 5)}form a basis for R? O Yes O No

A:

Q: Determine whether or not the given vectors form a basis of R4 V1 %3D (-2, 6, 2, 4), v2 3D (1,0,…

A: To Determine: whether or not the given vectors form a basis of R4 Given: we have  v1=-2,6,2,4…

Q: Write down a basis for the vector space L(R", R).

A: Given vector space is the finite dimensional vector space of dimension m.

Q: Find the dimension of the vector space V and give a basis for V. (Enter your answers as a…

A:

Q: Let {x, y} be a basis for a vector space V. Show that both {x + y, ax} and {ax, by} are bases for V,…

A:

Q: Consider the set P= {1+2x,-1-x+x*,x+x,1+2x+x}. i. Determine if the set P spans the vector space P, ,…

A:

Q: Find out if the given system of vectors is a basis of the four-dimensional Euclidean space R': 2 3…

A:

Q: Find out if the given system of vectors is a basis of the four-dimensional Euclidean space R*: 2

A:

Q: Write the standard basis for the vector space.M3,3

A: The set of  all nXn matrices with real entries is a vector space and is denoted by Mn,n . Standard…

Q: Consider the system of vectors (1,2, 1, 1)", (0,1,3, 1)", (0,3,2,0)", (0,1,0,0)". a) Prove that it…

A:

Q: Explain the term A Basis for a Vector Space.

A:

Q: Find an orthogonal basis for R3 spanned by the vector A= {(1,2,2),(-1,0,2),(0,0,1)}

A:

Q: Assume that the vector space R' has the Euclidean inner product. Apply the Gram-Schmidt process to…

A: The given vectors u1→=1,1,1, u2→=0,1,1, u3→=0,0,1

Q: Let W be the set of all vectors with x and y real. Find a basis of W-

A: Here, W is a set of vectors of the form xyx+y=x101+y011 with x and y are real numbers. Now, the…

Q: Let W be the set of all vectors with x and y real. Find a basis of W1 x + y,

A:

Q: Find a subset of the vectors that forms a basis for the space spanned by the vectors, then express…

A:

Q: Find a basis of the subspace of Rª that consists of all vectors perpendicular to both 1 1 and -6 2 마…

A:

Q: Find a subset of the vectors that forms a basis for the space spanned by the vectors, then express…

A: Answer:

Q: Let V be an n-dimensional vector space. Prove that any system {vi,...ūn} of linearly independent…

A: Let V be an n-dimensional vector space. We have to prove that any system v1→,v2→,v3→,.....,vn→ of…

Q: Let u and v be two vectors in R°. Determine the most general condition under which {u, v, u × v}…

A:

Q: Do the vectors form a basis for R"? Show which conditions, if any, are met, and show which…

A:

Q: Find the coordinates of (2, 3, 4, – 1) with respect to the basis B = {(1, 1, 1, 2), (1, –1, 0, 0),…

A:

Q: 2. Find a subset of the vectors that is a basis for the space spanned by these vectors. v₁ =…

A: Solution:-

Q: Find a basis for the orthognonal complement of U = span 1 -2 1 - 1 -2 1 Number of Vectors: 1 Basis…

A: Solution:

Q: Determine whether the set of vectors {(2,-1,3),(4,1,2),(8,-1,8)} form a basis for R3.

A: Consider the vectors : 2,-1,3, 4,1,2, 8,-1,8 To form a basis these vectors, the vectors must be…

Q: Find an prthonormal basis of the plane x1+ 2x2- x3 0.. W

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 3. Do the vectors (1, 1,0, 0), (1, -1,0,0), (0, 0, -1, 1) and (0, 0, -1, –1) form an orthogonal…

A: The solution is given as

Q: Show that if a vector space V has one basis of n distinct vectors, then every basis for V has n…

A:

Q: The vectors x1 = (1, 1, 1, 1), x2 = (0, 1, 1, 1), x3 = (0, 0, 1, 1), and X4 = (0, 0, 0, 1) form a…

A: Given the vectors:

Q: Let B, = {v,, v,,v, } be a basis for a vector space V. Determine whether or not B, = {ū,,ū,,ūz} is a…

A: Let V be vector space over field F . S⊂V(F) is said to be basis if S is Linearly independent Linear…

Q: Prove that a linearly independent system of vectors v1, V2, . .. , Vn in a vector V is a basis if…

A:

Q: Find an orthogonal basis for the vector space spanned by 0 and 2

A: To Determine :-  An orthogonal basis for the vector space spanned by  203   and   121.

Q: Show that B = {v1, v2, v3} is an orthogonal basis for R3: v1 = (3,−3,0), v2 = (2,2,−1), v3 =…

A:

Q: Find an explicit description of the null space of matrix A by listing vectors that span the null…

A:

Q: Show that the vectors u =, v= and w= form a basis for R3

A:

Q: Find a basis for the orthognonal complement of U = span 1 -1 -2 -2 -1 3 2 Number of Vectors: 1 0…

A:

Q: basis

A:

Q: a. Find the change of basis matrix from the basis B to the basis C. [id] = b. Find the change of…

A:

Q: Do the vectors form a basis for R"? Show which conditions, if any, are met, and show which…

A: First one not form a basis but second one is.. See the step 2

Q: In the vector space P, of polynomials of degree at most two, the coordinates of the vector…

A: The basis of the given polynomial is: B= e1=1+2x,  e2=x-3x2,  e3=1+x2

Q: Determine

A: Let,given set of vectors is V and V = {(1, 1, 0), (1, 0, 1), (0, 1, 1)}

Q: Explain how fixing a basis v1, , Vn of V associates to a quadratic form a corresponding function in…

A: To explain how a basis v1,v2,...,vn associates to a quadratic form of a corresponding function in n…

Q: Let B = {(-1,2, 1), (1,3,0), (1,0,0)} be a set of vectors in R. Let ū = (1,3,2). - Verify that B…

A:

Q: . Let W be the subspace of R' generated by the vectors (1, –2, 5, –3), (2, 3, 1, –4) an (3, 8, –3,…

A: Consider the metrix A with given three vector as row…

Q: Let u and v be distinct vectors of a vector space V. Show that if {u, v} is a basis for V and  a and…

A: Given,  u and v be distinct vectors of a vector space V and  {u, v} is a basis for V and  a and b…