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- (7) Given the following pieces information, describe the set of solutions x such that Ax (a) b is not in C(A). (b) b is in C(A), and there is a vector c which satisfies the property that the set of solutions of Ax = c is a plane. (c) A is a 6 x 7 matrix, b is in C(A), and N(A") is a four dimensional vector space. (d) A is a 3 x 3 matrix, and b is the first column of A. Additionally, Ax = b: O is solved by both x = and x =arrow_forwardIf T is defined by T(x) = Ax, find a vector x whose image under T is b, and determine whether x is - 6 1 – 4 2 unique. Let A=| 0 1 -3 and b=| - 17 4 - 17 8 Find a single vector x whose image under T is b. x = Is the vector x found in the previous step unique? A. No, because there are no free variables in the system of equations. B. Yes, because there is a free variable in the system of equations. C. Yes, because there are no free variables in the system of equations. D. No, because there is a free variable in the system of equations.arrow_forwardProblem 2. Let P3 be the vector space of all polynomials of degree 3 or less in the variable r. Let 2 - r + 2? 6 – 3x + 3x? - 3z, 2+ 2? – z', P1(z) P2(x) p3(2) P1(z) 3 - z + 2x? – 273 and let C = {p1(r), p2(2), pa(r), pa(z)}. a. Use coordinate representations with respect to the basis B = {1, z, r?, z'} to determine whether the set C forms a basis for P3. choose %3D b. Find a basis for span(C). Enter a polynomial or a comma separated list of polynomials. { } c. The dimension of span(C) isarrow_forward
- -7 4 -4 4 Let T = 3 -7-4-1 -6 A nonzero vector with integer entries in the null space of T'is 00000arrow_forward3. Suppose A = 3 1 1 -2 (a) Find a vector X E R² such that AX = (b) Find a vector X € R² such that AX = B 9arrow_forward3. Consider the linear operator T: R2 R² defined by the following equations. W1 = 3x1 +11xz W2 = X1 +7x, Find T-1(w1, w2).arrow_forward
- 1. (a) Find the matrix P that multiplies every vector (x, y, z) E R³ to produce the vector (3x+2y+z, 5y- z, 8x). Also find P-1. (b) Find the matrix P that multiplies every vector (x, y) E R2 to produce (5x-4y, - 2x, 3y - 2x) = R³arrow_forward6. If s is a scalar value and s + (ỷ xw) ( x w) where and w are two vectors in R, then s+ (3 x w) (3 x w) is a. a vector in R³ b. a scalar C. undefined d. a 3 by 3 matrix. Clearly explain your answer.arrow_forward1. Show that the given vector functions are linearly independent on (-∞, ∞). 2t H 0 x₁ (t) = X1 t 0 -t3 2 X₂ (t) = 3t² 2 x3(t) = t 0 3+3arrow_forward
- Q.1. The set of all positive real numbers with the operations x + y = xy kx = xk Is it a vector space? Justify your answer.arrow_forwardIf T is defined by T(x)=Ax, find a vector x whose image under T is b, and determine whether x is unique. Let A= (see picture) and b= (see picture). Is the vector x found in the previous step unique?arrow_forwardby Problem 2. Consider the set V = R² with addition and scalar multiplication defined (X1, X2) ℗ (Y1, Y2) = (X1 + X2, Y₁ + y2), a (x₁, x₂) = (ax₁, x₂). Is V a vector space with these operations? Justify your answer.arrow_forward
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