Question

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Let V be the vector space of 2 × 2 real matrices and let P2 be...

Let V be the vector space of 2 × 2 real matrices and let P2 be the vector space of polynomials of degree less than or equal to 2. Write down a linear transformation T : V ? P2 with rank 2. You do not need to prove that the function you write down is a linear transformation, but you may want to check this yourself. You do, however, need to prove that your transformation has rank 2.

Solutions

Expert Solution

Let A =

a	b
c	d

be an arbitrary 2x2 matrix in V where a, b, c, d are arbitrary real numbers. Also, let p(x) = ax²+bx be an arbitrary polynomial in P₂. Now, let us define T : V ? P₂ by T(A) = p(x).

Further, let {E₁₁,E₁₂,E₂₁,E₂₂} denote the standard basis of V where E_ij is the 2x2 matrix with 1 as the ijth entry, the remaining entries being 0. Also,{ 1,x,x²} is the standard basis of P₂.

Now, T(E₁₁)= x²,T(E₁₂)=x,T(E₂₁)=0 and T(E₂₂)=0. Then the standard matrix of T is [T(E₁₁),T(E₁₂),T(E₂₁),T(E₂₂)] = M =

1	0	0	0
0	1	0	0
0	0	0	0

It may be observed that the entries in the columns of M are the coefficients of x² and x in T(E₁₁), T(E₁₂), T(E₂₁) and T(E₂₂) respectively.

It may also be observed that M has 2 non-zero rows so that rank(M) = 2. Therefore, rank(T) = rank(M) = 2.

Let A =

a	b
c	d

and B =

e	f
g	h

be 2 arbitrary matrices in V and let k be an arbitrary scalar. Then A+B =

a+e	b+f
c+g	d+h

and kA =

ka	kb
kc	kd

so that T(A+B) = (a+e)x²+(b+f)x = ax²+bx+ ex²+fx = T(A) +T(B). Thus, T preserves vector addition.

Also, T(kA) = kax²+kbx = k(ax²+bx) = kT(A). Thus, T preserves scalar multiplication. Hence T is a linear transformation.

milcah answered 1 year ago

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