MCQ1: Let w and v be vector spaces. A function T: V → W is called a linear transformation from v to w  if for all x    and scalar K I.                       =   Answer: I and II FBQ1: Let T:U→V be a linear transformation, defined  btTU=0 ∀  u∈U. Then we call T a __________________________ Answer: Null zero transformation FBQ2: If U and V are two vector spaces over a field F and T:U→V  is a bijection linear transformation. Then we say U and V are …………………………………..  Answer: Isomorphic FBQ3: Another name for linear transformation is …………………………………………… Answer: Vector space homomorphism FBQ4: The rank of a linear transformation T is defined to be ………………………………….  Answer: Dimension R(T) FBQ5: Suppose U is a vector space over a field F, and T is an identity transformation, then the function T:U→U will be defined by __________________________________  Answer: T(u)=U FBQ6: A homomorphism theorem states that if v and w are vector spaces over a field F and T:VT:V→W is a linear transformation. Then VKer T _______________________ Answer: R(T) FBQ7: The nullity of T= ___________ Answer: Dimension of ker T FBQ8: A linear transformation T:U→U is called _________________ if each v∈V, there exists u∈U such that Tu=v that RT=V. Answer: Subjective FBQ9: Two finite –dimensional vectors U and V are isomorphic if and only if ________________ Answer: Dimension of U= Dimension of V FBQ10: Let U,V be vector spaces over a field F of dimension m and n respectively, then L(U, V) is a vector of dimension _______ Answer: Mn FBQ11: L(R2, R) is a real vector space of dimension _______ Answer: 2 FBQ12: Let U be a vector space over F, then the space L(U, F) is called the _________of U Answer: Dual space FBQ13: A transformation on T:U→F is called __________ Answer: Linear function FBQ14: The basis f1,f2, …, fm of V is called the ___________ of the basis e1,e2,…, em of V Answer: Dual basis FBQ15: A polynomial Px=a0+a1x+…+an-1xn-1+xn is called ________ Answer: Monic polynomial FBQ16: A … is a sequence in which each successive terms of the sequence are in equal ratio. Answer: geometric progression FBQ17: For T∈AV, the unique monic polynomial P of the smallest degree such that PT=0 is called __________ T Answer: Minimal polynomial FBQ18: The division algorithm states that given f(x) and p(x), there exist polynomial g(x) and h(x) such that __________ hx=0 Answer: F(x)=p(x)g(x)+h(x) FBQ19: For any vector space V, the minimal polynomials for the identity transformation and the zero transformation are x-1 and ______ Answer: X FBQ20: Every vector space is isomorphic to its ______________________ Answer: Second dual FBQ21: The degree of the polynomial (x2-1) is __________________ Answer: 4 FBQ22: The matrices are said to be equal if they are of the ____________ Answer: Same order and element FBQ23: A square matrix A such At=A is called a ________ Answer: Symmetric matrix FBQ24: A square matrix A such At=-A is called a ________ Answer: Anti- symmetric matrix FBQ25: A matrix obtained by replacing each of its entry by complex conjugate is called …………………….  Answer: Conjugate matrix FBQ26: In conjugate matrix, A-=A- if only if A is a called __________ Answer: Real matrix FBQ27: Given a matrix A∈Mm×nF, the matrix formed by taking conjugate of matrixAt is called _________ Answer: Conjugate transpose of A FBQ28: A square matrix A for which A-t is called _______________ Answer: Hermitian matrix FBQ29: A square matrix A for which A-t=-A is called _______________ Answer: Skew- Hermitian matrix FBQ30: The conjugate of 1123 is ……………………………………  Answer: Its self FBQ31: For a real matrix A, A is Hermitian if A is ……………………………. Answer: Symmetric FBQ32: For a real matrix A, A is skew-Hermitian if A is ……………………. Answer: Skew- symmetric FBQ33: A matrix whose entries along the diagonal are non-zero is called …………………….  Answer: Diagonal matrix FBQ34: A square matrix A∈MnF is said to be _______________ if there exists B∈MnF such that B=BA= In Answer: Invertible FBQ35: The integer PcA=PrA is called ___________ of A, and is denoted by PA. Answer: Rank FBQ36: A matrix obtained by subjecting 1n to an elementary row or column operation is called _________ Answer: Elementary matrix FBQ37: A m×n matrix A with the following properties (i) The non-zero rows come before the row(ii) In each non-zero row, the first non-zero entry is 1.(iii) The first non-zero entry in every non-zero row ( after the first row) is to the right of the first non-zero entry in the preceeding row is called…………………….. Answer: Row-reduction echelon matrix FBQ38: If E is a row-reduction echelon form of A. Then, the rank of A is …………………….  Answer: Number of non-zero rows of E FBQ39: Consider a matrix A=2513, its determinant is _________ Answer: 1 FBQ40: The determinant rank of m×n matrix A is equal to the ___________ Answer: Rank of A FBQ41: The rank of A=1425 is ________ Answer: 2 FBQ42: If A=126541732,then, the determinant of A is _________________ Answer: -13 FBQ43: If A is a linear transformation represented by a matrix A and there is a vector X∈Rn ≠0 such that AX=λX, for some scalar λ, then is called __________ Answer: Eigen value FBQ44: For an eigenvalue λ of T, the non-zero subspace W λ is called the ________ of T associated with eigenvalue. Answer: Eigen value FBQ45: The eigenvalue for the linear operator T:R3→R3 such that Tx, y,z=2x, 2y,2z is _______________ Answer: 2 FBQ46: A linear transformation T:V→V on a finite dimensional vector V is said to be _____if there exist a basis B=v1,v2, …,vn of V such that the matrix of T with respect to the basis B is diagonal. Answer: Diagonalisable FBQ47: The __________________ of a matrix A over F is the monic polynomial p(t) such that (i) PA=0 and (ii) if q(t) is non-zero polynomial over F such that degq<degp, then qA≠0. Answer: Minimal polynomial FBQ48: The determinant of A=sinθ-cosθcosθsinθ is _______________________ Answer: 1 FBQ49: If A=1020, then PrA is ___________ Answer: 2 FBQ50: If B=-iiii, where i is a complex value, then |B|2 is ­­­­____________ Answer: 4 MCQ1: Let w and v be vector spaces. A function T: V → W is called a linear transformation from v to w if for all x x,y∈V and scalar K T(x+y) = Tx+T(y) TKx=KT(x) Answer: I and II MCQ2: Which of the following is linear Answer: F: R→ R defined in by fx=2x MCQ3: Which of the following is not a linear transformation? Answer: None of the options MCQ4: Given a linear transformation T: U→ V, which of the following is true?  Answer: All of the options MCQ5: Which of the following is true for this linear transformation T: U→ V is one – one if and only if kerT = (0) onto if and only R(T)=V   Answer: I and II MCQ6: Two finite-dimensional vectors space U and V are isomorphic if and only if Answer: Dim U = dim V MCQ7: In the rank unity theorem, Dim V – nullity (ST) = DIM V – nullity (T) – DIM R((T) ∩ kerS) which implies  Answer: Nullity (ST) = nullity (T) + dim (R(T) ∩ kerS) MCQ8: The minimal polynomial of a matrix A over f is the monic polynomial P(t) such that I.P(a) = 0II.If q(t) is a non-zero polynomial over F such that deg q < deg p, q(A) ≠ 0 Which of the following is property of minimal polynomial? Answer: I and II MCQ9: If the characteristic polynomial T:R4→R4 is (t+1)2(t-2)2, then the minimal polynomial could be  Answer: (t+1)(t+2) MCQ10:  What is the characteristic polynomial of A if 211-12-1-113 Answer: ft=t3-7t2+19t-19 MCQ11:  Let 32-10, then the characteristic polynomial of A is Answer: t2-3t+2 MCQ12: Let T:V→V be a linear transformation. A vector x∈V is an Eigen vector of the linear transformation T ifX is none zeroTx=ʎx for some scalar ʎ∈F. Which of the following is the definition of eigen vector?   Answer: I and II MCQ13:  Obtain an eigen value for the linear operator T:R^3→R^3 by T(x,y,z)=(2x,2y,2z)  Answer: 2 MCQ14: Two matrices are said to be equal if I.They have the same size. i.e, they have the same numbers of rows as well as columnsII.Their elements at all the corresponding positions are the same.Which of the following qualify the definition of equal matrices?  Answer: I and II MCQ15: Find the eigen values of 2221  Answer: 0, 3 MCQ16:  Describe T:R3→R3 such that T]B = 124231312, where B is the standard basis of R3  Answer: Tx,y,z=(x+2y+4z, 2x+3y+z, 3x+y+2z)  MCQ17: Calculate 312+ 01   Answer: 39   MCQ18: If A is an upper triangular 3×3 matrix, say A = 123045006. Therefore At is   Answer: Lower triangular MCQ19: A matrix A is invertible when Answer: The determinant is zero MCQ20: Let A = 100708009 , B = 213540 , find AB if it is defined   Answer: 21467360   MCQ26: Let U, V, W be vector spaces over F. Suppose S∈L(v,w) and T∈L(u,v), then we have   Answer: S₀T∈L(u,w) MCQ27: Let T:R2→R3 and S:R3→R2 be defined by Tx1,x2=x1,x2,  x1+x2 and Sx1,x2,x3=x1,x2. Then one of the following is true  Answer: S₀T≠T₀S MCQ28:  The required polynomial for any vector space V, the minimal polynomial for identity I and 0 the zero transformation is  Answer: x-1and x MCQ29: The sum of matrix A and B where B is the identity matrix with respect to addition will give the matrix  Answer: Matrix 0 MCQ30:  In properties of matrix addition, the equation A + B = B + A refers to Answer: Commutative MCQ31: The transpose of 2 by 3 matrix will give a Answer: 3 by 2 matrix MCQ32: Let [aij] be a square matrix, then the entries a11, a12,a13, …,a1n are called  Answer: The diagonal entries of A MCQ33: The conjugate of (2 3+i i ) is  Answer: (2 3-i -i ) MCQ34: For a matrix A = 1220, we have the following except  Answer: A=AT MCQ35: Find det(T) where we defined T : R3→R3 by T(x1, x2, x3) = (3x1 + x3, -2x1 +x2,-x + 2 ×2 + 4x3)  Answer: 9 MCQ36: Obtain the cofactor C12 of the matrix A = 02-1341216 Answer: -16 MCQ37: Given A = 102310001 and B = 2109038005. Calculate ꞁABꞁ   Answer: 30 MCQ38: If A = 100120, find Pr (A) Answer: 2 MCQ39: Let T : R2→R2be defined by Tx,y=(x,-y) for all x,y∈R. Show that T is a linear transformation  Answer: T(x1 + y1) +β(x2 + y2)) =αT(x21 + y1) + βT(x2 + y2)  MCQ40: If Let T:U→V is one - one and onto linear transformation, then we can have Answer: T-1:V→U MCQ41: Obtain the determinant rank of A=1425  Answer: 2 MCQ42: Obtain the characteristic polynomial of the matrix 120-1  Answer: t2-1 MCQ43:  The minimal polynomial of A=02-1341216 is either  Answer: (t-1)(t-2) or (t-1)(t-2) MCQ44: Let U and V be finite dimensional vector space over F and T:U→V be a linear transformation, then rank (T) + nullity (T) = ?   Answer: dim U MCQ45: Let T:U→V be a linear transformation, then Tis 1-1 . if T(U1) = T(U2) implies that  Answer: U1= U2  MCQ46: A matrix having three horizontal rows and four vertical columns is called   Answer: 4 by 4 matrix MCQ47:  If 1023=xyz3, find x, y and z Answer: x=1, y=0,z=2 MCQ48: What is the sum of 1001 and -100-1  Answer: 0000 MCQ49:  Calculate 2B, where B=121413000 Answer: 11223000 MCQ50: Calculate 312  Answer: 36 MCQ21: Let e1=0,1,0 and e2=0,0,1 form the standard basis of R3. Let 1,2,2,3 and 3,4 be three vectors in R2. Obtain the linear transformation T:R3→R2 such that T(e1)=1,2, T(e2)=2,3 and T(e3)=3,4  Answer: Tx1,x2,x3=(x1+2x2+3x3,  2x1+3x2+4x3) MCQ22: Given T:U→V is one – one if and only if Answer: KerT = (0) MCQ23: Given a linear transformation T:U→V is onto if and only if Answer: RT= kerV MCQ24: Let S, T ∈L(u,v) where S and Tare linear transformation. We define S+T:U→V by (S+T)U=  Answer: Su+T(u) MCQ25:   Answer: