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- x 2y 3. Let T ((1)) = (1 + 2) 4x a) T can also be written as T b. matrix you found above. (Ⓒ)) = ¹ (*) A c) Use your answer to b) to find a vector (Ⓒ) for some matrix A. What is the matrix A? ·((*)). such that T || = d) Use your answer to b) to find a solution to the system of equations x + 2y = = 2 4x + 9y = -1arrow_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_forward(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_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. Consider the linear operator T: R2 R² defined by the following equations. W1 = 3x1 +11xz W2 = X1 +7x, Find T-1(w1, w2).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
- 3. 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_forward1. (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_forwardProblem #4: Let p = 7x² + 3x + 4. Find the coordinate vector of p relative to the following basis for P2, P₁ 1, P2 = 1 + x, P3 1+x+x². =arrow_forward
- Let c₁ (t) = e5¹i + 7 sin(t)j + tªk and c₂(t) = e¯5¹i + 3 cos(t)j – 4tªk. (Enter your solution as a single vector using the vector form (*,*,*). Use symbolic notation and fractions where needed.) d dt −[c₁ (t) + c₂ (t)] = =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_forwardProblem 5. The functions f(x) = 2x and g(x) = |x| can be considered elements of the vector space C[a, b] for any interval [a, b] CR. Show that f(x) and g(x) are linearly independent in C[-1, 1] while being linearly dependent in C[0, 1].arrow_forward
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