8/11/2019 Tutorial 2 Mal101

1/1

13

MAL101:: Tutorial 2 :: Linear Algebra

Notation: F = Ror C, Pn:= {f F[x] : deg f < n}

(1) Suppose v1= (1, 2) v2= (0, 1) R2.

(a) Describe geometrically the subsets W1:= {tv1: t R}, W2:= {tv2 : t R},W3 := {sv1+tv2: s, t R} and W4 := {sv1+tv2: 0 s, t 1}.

(b) Which ofW1, W2, W3, W4 are subspaces ofR2? Justify your answer in each case.

(c) Show that {v1, v2}is a linearly independent subset ofR2.

(d) Suppose v3= (2, 3). Is{v1, v2, v3} linearly independent?(2) Suppose V := C2 is the complex vector space (over C) under component-wise addition.

(a) Show that {(1 + i, 2), (2, 1)} is linearly independent.(b) Show that {(1, 2), (0, i), (i, 1 i)} is linearly dependent.(c) Show that every ordered pair can be written as a linear combination ofv1 = (1 + i, 2) and

v2= (2, 1). Also show that up to change of order (ofv1 andv2) such a linear combinationis unique (for each ordered pair).

(d) Show that every ordered pair can be written as a linear combination ofv1 = (1, 2), v2 =(0, i), v3= (i, 1 i) in more than one ways.

(3) Show that X = {(1 +i, 1 i), (1 i, 1 +i), (2, i), (3, 2i)} is linearly independent in C2(R).Express (a +ib, c+id) as an R-linear combination of vectors belonging to X.

(4) Let Vbe a vector space over F. Show that u, v ,w Vare linearly independent if and only ifu+v, v+w, w+u are linearly independent.

(5) (a) Find the coordinates of (a,b,c) R3

relative to the ordered basis {(1, 0, 0), (1, 1, 0), (1, 1, 1)}.(b) Find the coordinates ofa+bX+ cX2 relative to the ordered basis {1, 1 +X, 1 +X2} inthe spaceP3 of polynomials of degree at most 2 with coefficients from R.

(c) Find the coordinate vectors of an element ofR3 with respect to the following bases B1 ={(1, 2, 1), (1, 2, 3), (0, 1, 1)} and B2= {(1, 0, 0), (1, 1, 0), (1, 1, 1)}. Also write the change ofcoordinate matrix.

(6) (a) Show that ifv V then Fv:= {v: F} is a subspace of any vector space V over F.(b) Show that ifW1, W2 are subspaces ofV, then W1 W2 is a subspace ofV.(c) Show that the intersection of any collection of subspaces of a vector space is a subspace.(d) SupposeW1 andW2 are subspaces of a vector space V. Show that W1 W2 is a subspace

ofV if and only if either W1 W2 or W1 W2.(e) LetXbe a nonempty subset of a vector spaceV over F. Letspan(X) :={

n

i=1aivi: n

N, ai F, vi X}and let Xbe the intersection of all the subspaces ofVwhich containX. Show that span(X) andXare subspaces ofV. Also show that span(X) = X.

(7) In each case show that W1+ W2 = V (directly) and find dim(W1 W2). Verify the formuladim(W1+W2) = dimW1+ dimW2 dim(W1 W2).(a) V = R2, W1 is the X-axis,W2 is the Y-axis.(b) V = R2, W1 andW2 are distinct lines through the origin.(c) V = R3, W1 is the XYplane and W2 is the Y Zplane.(d) V = Mn(R), W1 ={A Mn(R) : A is upper triangular }, W2 :={A Mn(R) : A is

lower triangular}.(e) V = Mn(R), where W1 is the space ofn n symmetric matrices and W2 is the space of

n n skew-symmetric matrices.(8) Which of the following is a linear transformation? Justify.

(a) T1: R2 R2 defined by T1(x, y) = (x

2 +y2, x y) over R.(b) T2: R

2 R2 defined by T2(x, y) = (x +y+ 1, x y) over R.(c) T3: R

2 R2 defined by T3(x, y) = (ax +by, cx+dy) over R.(d) T4: C Cdefine by T(z) = zover C.(e) R2 to R2 the rotation about the origin by an angle . (Write an expression for rotation.)(f) T5: Mmn(F) Mnm(F) defined by T5(A) =A

t (At is the transpose ofA).(g) T6: Mn(F) Fdefined by T6(A) = tr(A).(h) T7: Pn Pn such that T7(p)(x) = p(x 1).(i) T8: Pn Pn+1such that T8(p)(x) = xp(x) +p(1).

::: END :::