In: Math

# 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 Eij 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.

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