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Let V be the vector space of polynomials of degree 3 or less in the variable (x) with real coefficients. Note that (mathcal{B}_1 = {v_1 = 1, v_2 = x, v_3 = x^2, v_4 = x^3}) is a basis for V. Let W be the vector space of polynomials of degree 4 or less in the variable (x) with real coefficients. Note that (mathcal{B}_2 = {w_1 = 1, w_2 = x, w_3 = x^2, w_4 = x^3, w_5 = x^4}) is a basis for W. For each vector space define an inner product by (langle f(x), g(x) angle = int_{-1}^{1} f(x)g(x) dx.) Let (T: V o W) be defined by (T(f(x)) = xf(x)) for all (f(x) in V). Note (you do not have to prove this) that T is a linear transformation. 1. (2 pts.) Find a matrix that corresponds to T using the bases (mathcal{B}_1) and (mathcal{B}_2). Explain. 2. (6 pts.) Find a matrix that corresponds to the adjoint (T^*) using the bases (mathcal{B}_2) and (mathcal{B}_1). Prove that your matrix is correct.

          Let V be the vector space of polynomials of degree 3 or less in the variable (x) with real coefficients. Note that (mathcal{B}_1 = {v_1 = 1, v_2 = x, v_3 = x^2, v_4 = x^3}) is a basis for V. Let W be the vector space of polynomials of degree 4 or less in the variable (x) with real coefficients. Note that (mathcal{B}_2 = {w_1 = 1, w_2 = x, w_3 = x^2, w_4 = x^3, w_5 = x^4}) is a basis for W. For each vector space define an inner product by (langle f(x), g(x)
angle = int_{-1}^{1} f(x)g(x) dx.) Let (T: V 	o W) be defined by (T(f(x)) = xf(x)) for all (f(x) in V). Note (you do not have to prove this) that T is a linear transformation.

1. (2 pts.) Find a matrix that corresponds to T using the bases (mathcal{B}_1) and (mathcal{B}_2). Explain.

2. (6 pts.) Find a matrix that corresponds to the adjoint (T^*) using the bases (mathcal{B}_2) and (mathcal{B}_1). Prove that your matrix is correct.

Let V be the vector space of polynomials of degree 3 or less in the variable (x) with real coefficients. Note that (mathcalB1 = v1 = 1, v2 = x, v3 = x^2, v4 = x^3) is a basis for V. Let W be the vector space of polynomials of degree 4 or less in the variable (x) with real coefficients. Note that (mathcalB2 = w1 = 1, w2 = x, w3 = x^2, w4 = x^3, w5 = x^4) is a basis for W. For each vector space define an inner product by (langle f(x), g(x)
angle = int-1^1 f(x)g(x) dx.) Let (T: V o W) be defined by (T(f(x)) = xf(x)) for all (f(x) in V). Note (you do not have to prove this) that T is a linear transformation.

1. (2 pts.) Find a matrix that corresponds to T using the bases (mathcalB1) and (mathcalB2). Explain.

2. (6 pts.) Find a matrix that corresponds to the adjoint (T^*) using the bases (mathcalB2) and (mathcalB1). Prove that your matrix is correct.

Added by Scott M.

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