Let L be a linear operator on Rn. Suppose that L (x) = 0 for some x ≠ 0. Let A be the matrix representing L with respect to the standard basis {e1, e2, . . . , en}. Show that A is singular.

Q: Let B = {1, x, x², x*} be an ordered basis for P3. Find the coordinate vector of f(x) = 4x3 – 72² +…

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Q: Given the coordinate matrix of x relative to a (nonstandard) basis B for R, find the coordinate…

A: We will find the coordinate of vector x with respect to the standard basis as follows

Q: Consider polynomials with degree less than or equal to n. We can think that each polynomial…

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Q: Let V be a vector space and suppose that T is a linear operator on V such that T(2u-v)= 3u+5v and…

A: T is a linear operator on V,T2u-v=3u+5vT-2u+2v=-4u-6v

Q: Let A be an m x n matrix with columns C₁, C₂,, Cn. If rank A = n, show that {ATc₁, AT c₂,, ATc₂} is…

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Q: Let A be an m × n matrix and u a vector in Rn. Suppose that Au = 0 (where 0 is the zero vector in…

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Q: Let CE Rxn be a matrix such that xCx>0 for all nonzero vectors x ER". Prove that C is nonsingular.

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Q: Recall P := {a + bt : a, b E R}. Let T : Pf → PR be the derivative operator, i.e. T(p(t)) := p'(t).…

A: Given: ℙ1ℝ=a+bt : a, b∈ℝ and T:ℙ1ℝ→ℙ1ℝ is defined as T(p(t))=p'(t). A is matrix of T with respect to…

Q: Let V be a vector space with dim(V) = 3. Explain why any T: V → V has at least one real eigenvalue.

A: We know that the eigenvalues are zeros of the characteristic Polynomial of a linear transformation.…

Q: Let T be the linear operator on R 2 defined by T(x, y) = (−y, x) i. What is the matrix of T in the…

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Q: = 4. Let T₁(x) = A₁(x) and T₂(x): = A₂(x) be defined by the following matrices A₁, A2. Let T = T₂ 0…

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Q: Let B = {v1, .. be a linearly independent set in Rn. Explain why B must be a basis for TR"

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.