Deepikachaudhary posted an Question

July 10, 2020 • 14:22 pm 30 points

V then w is c is a vector space over f with respect to the operations of vector addition and scala improper and proper subspaces: if v is any vector space then

V Then W IS C is a vector space over F with respect to the operations of vector addition and scala Improper and Proper Subspaces: If V is any vector space then V itself is Ler Vbe a vector space over the field F and W Vector Subspaces zero vector alone i.e. {0) is a subspace of V always So and V-Tmproparu The subset consisting of the zero vector alone i.e. {0} is a subspace of V always oper subspaces. The subspaces SuDspace of V. These two subspaces are called impro V are called proper subspaces VThese two. fV Tests for Showing Subspaces Wo-step Test : A subset W of a vector space VF) is a sub space of W is non-empty VX, y E W X - y e W va e Fx e W ax e W 1 2. eP lest : A non-empty subset W of a vector space V(F IS a suD Example: Which of the following's is/are subspaces of i.e. {(x.y, z)|x, y, Z e R (R) By e W. Va,ß e F and Vx, y e W R( (1) W ={(x, y, z) /ax + by + Cz = 0 for given a,b,c E R (2) W ={(x, y,z) |3x +4y = 7z} (3) W ={(%.y.0) |x,ye nce.co W =(%y,2) |y e z} W=(xy.2) ky = 0} W=(x.y,z) |x+y +Z=1 W ={%.y,2) x+y+zeZ W= {(xy.z) x.y,zeN} (4) (5) (6) (7) (8)

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