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Differential Equations and Linear Algebra (4th Edition)
- Consider the matrices below. X=[1201],Y=[1032],Z=[3412],W=[3241] Find scalars a,b, and c such that W=aX+bY+cZ. Show that there do not exist scalars a and b such that Z=aX+bY. Show that if aX+bY+cZ=0, then a=b=c=0.arrow_forwardThe space P₂ represents all 2nd degree or less polynomials. A polynomial such as p(x) = 1 + 4x + 6x² 4 in P₂. The standard basis polynomials for this space are 6 would be represented as the vector {1, x, x²}. 3 The function F, defined by F(p(x)) = [₁ p(x 3 a) What should be the size of the matrix representing the function F? Answer: a = b) Write the matrix for this linear transformation according to the standard basis polynomials. [Hint: Find where the standard basis polynomials go under this transformation.] b c) Find two different non-zero polynomials in the null space of this transformation. First polynomial: a + bx + cx² where: C = rows and a = Second polynomial: a + bx + cx² where: b = p(x) dx, is a linear transformation from P₂ to R. columns. C = Null space?arrow_forwardConsider the following ordered basis for P₂ (R): B = {x² + 2x + 1, x + 1, 1} (a) The quadratic polynomial that has [[1], [3], [2]] as its coordinate vector relative to B is: (b) The coordinate vector (relative to B) of x² + x + 1 is:arrow_forward
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