Linear Algebra Review Session 3

Linear Algebra: Review Session 3 Matrices Matrix of a Linear Transformation and Coordinate Vectors Ex. 1. Let M22(R) be the vector space of 2 x 2 matrices with real coefficients and let -1)} be its standard banks. (a) Let T: M2x2(R)+Max(R) by T(A) A' where A' is the transpose of the matrix A. Compute the matrix of T with respect to the basis o. Is 7 onesto one? Is Tonto? (b) Let T: Max2(R)→R by T(A) (2) 1=a+b+c+d where A = Compute the matrix respect to the bases o for M2x2(R) and (1) for R. Is T one-to one? Is Tonto? of T with Ex. 2. Let P, be the vector space of polynomials with real coefficients of degree at most n. T: PP by T()=(1) d Define (a) Prove that T is linear. (b) Compute T(ar²+bx+c) where a,b,ce R. (c) Compute the matrix of T with respect to the standard bases (1,2,2) of P and (1. z.) of P 11111 Ex. 3. Let T: R R be the linear transformation with matrix representation T with respect to the standard bases on R5 and R³. Find a basis for Ker(T) and Im(T). Invertible Matrices and Invertible Maps 55777 Ex. 4. Compute the inverse of the matrix A 2 Ex. 5. Consider the linear transformation T: RR defined by T(..) (2x+y-42, 4y5c.). ). Determine whether or not T is invertible, and if so, find a formula for Ty Ex. 6. (a) Let U and W be vector spaces. Explain what it means to say that a linear transformation S: UW is invertible. (b) Let T:VV be a linear transformation and let that the matrix of T with respect to o 15 (1) be a basis for the vector space V. Suppose Explain how you know that T is invertible. 2 (c) Calculate T(t). Write your answer as a linear combination of the vectors and us Change of Basis Ex. 7. Let T R R be a linear transformation such that T 17(3)-(3) = and T (4)-(3) (a) Find the matrix of 7 with respect to the standard basis of R (b) Is T one-to-one? Is T onto? Justify your answers Ex. 8. Let P₂ be the vector space of polynomials with real coefficients of degree at most 2, and let a = (1,2+1,2x+1). It is a fact, which you may assume, that o is a basis for V. Suppose that 23 30-1 251 T:VV is a linear transformation and the matrix of T relative to o is T(3²+2+2). Find Ex. 9. Let P=(a+b+ct: a,b,cR) that T is linear. and T: P be defined by T(p) = →R p(1)] p(1) You may assume (a) Find the matrix representation of T with respect to the bases (1. f. 2) and {[]} (b) Find the rank and nullity of T. Einear Algebra Review Session 3: Matrices Ex. I Let a=E., Ez, Es, E3 a) T(Ei) = (8) = E. t →[T(c)]=[] [T] 100 0010 0100 ODOI T(E2)=(0)=(98)=3[]-[] t = [] T(E3)= (8) = (8) = EL is the matrix of t wrt & T(Ex)=(89) [T(Ex)]=[] t Rov-reducing, 000 we have y pinot calumas rank (T)= 4. RZAR3 0001 Since TM 2x2+Mzxz and dim (M2x2) =4, T is onto, Since TM2x2+M2x2 and T is onto, T is one-to-one ) T: M2x2 (R)+IR by T (A) = a+b+c+d b as well. T(E)=!~ [T(G)]=[1] : etc. => [T][1111] is the matrix of T urt doulß Now rank (T) =1 and dim(R) =1, so I is onto, But dim (Maxez (R)) =4 Daldun is greater than dim(k)=1, so I is not onto to-one. Ex.2 TP2P3 by T/F)= $fitidt. a ) let piftPz and coR. T(epte) = f(p+q) (eldt = of pro) + + fqdt = (p)+(4)、 X p(t) So, Tis linear. b) T(ax²+bx+c) = f(at²+bt tc) dt = jat³ + { bt² + ct/ = jax² + + bx² +cx c) Let 2=21,xx²3 and p=21,x,x², x's. ((1))=(0,1,0,0) [1] TC1 = x T(x)=x² T(x²) = x² 00 1 00 00 003 [T(0)=(0,0,2,0) ((x)](0,0,0) Example 3: Given: A matrix representing a linear transformation T. Task: Find a basis for the image (Im(T)) and kernel (Ker(T)) of T. Solution: Row-reduce the matrix to reduced row echelon form (RREF). Identify the pivot columns. Use the corresponding columns of the original matrix as a basis for Im(T). Solve the homogeneous system of equations associated with the RREF to find a basis for Ker(T). Example 4: Given: A matrix A. Task: Find the inverse of A (if it exists). Solution: Perform row operations on the augmented matrix [A | I] to obtain [I | A^-1]. If this is possible, then A is invertible and the resulting matrix is its inverse. Example 5: Given: A linear transformation T defined by its matrix representation. Task: Determine if T is invertible and find its inverse if it exists. Solution: Calculate the determinant of the matrix representing T. If the determinant is nonzero, then T is invertible. Find the inverse using the formula A^-1 = (1/det(A)) * adj(A), where adj(A) is the adjugate of A. Mathematical Expressions: Matrix representations: [1 3 3 3], [1 1 1 0 0], [0 0 0 1 1], etc. Row operations: R1 + R2, R3 - 2R1, etc. Determinants: det([1 3 4 1]) = -870 Vector notation: (1/3, 1/7), (x1, x2, x3), etc. Equations: x1 + x2 = 0, x3 + x4 + x5 = 0, etc. Example 6: Given: A linear transformation T:U -> W. Task: Determine if T is invertible and find its inverse if it exists. Solution: Define the inverse transformation S:W -> U. Verify that ST(u) = u for all u in U and TS(w) = w for all w in W. If such an S exists, then T is invertible and S is its inverse. Example 7: Given: A linear transformation T:R^2 -> R^2 defined by its matrix representation. Task: Determine if T is invertible and find its inverse if it exists. Solution: Calculate the determinant of the matrix representing T. If the determinant is nonzero, then T is invertible. Find the inverse using the formula A^-1 = (1/det(A)) * adj(A), where adj(A) is the adjugate of A. Example 8: Given: A linear transformation T:P2 -> P2 defined by its matrix representation. Task: Determine if T is invertible and find its inverse if it exists. Solution: Calculate the determinant of the matrix representing T. If the determinant is nonzero, then T is invertible. Find the inverse using the formula A^-1 = (1/det(A)) * adj(A), where adj(A) is the adjugate of A.