Answer to Question #305183 in Linear Algebra for misfit

Question #305183

Find the dimension and a basis for 𝑊 = {(𝑥, 𝑦, 𝑧, 𝑤,𝑡): 𝑥 + 𝑦 + 𝑧 + 𝑤 + 𝑡 = 0, 𝑥 − 𝑦 + 𝑧 − 𝑤 + 𝑡 = 0}


1
Expert's answer
2022-03-04T09:34:11-0500

WW is a subspace of R5\mathbb{R}^5. From equations x+y+z+w+t=0,xy+zw+t=0x+y+z+w+t=0,x-y+z-w+t=0 we find: t=wz+yxt=w-z+y-x . Substitute it in the first equation and get: x+y+z+w+wz+yx=2y+2w=0x+y+z+w+w-z+y-x=2y+2w=0 . Thus, we get: t=zxt=-z-x, w=y.w=-y. Thus, the space is 3 dimensional. x,y,zx,y,z are independent coordinates and coordinates tt and ww can be expressed via x,y,zx,y,z . To find the basis we take the standard basis from R3\mathbb{R}^3: x1=1,y1=0,z1=0x_1=1,y_1=0,z_1=0, x2=0,y2=1,z1=0x_2=0,y_2=1,z_1=0 and x3=0,y3=0,z3=1x_3=0,y_3=0,z_3=1 and extend it to R5\mathbb{R}^5 by using equations for tt and ww. We receive 3 vectors: e1=(1,0,0,0,1),e2=(0,1,0,1,0)e_1=(1,0,0,0,-1), e_2=(0,1,0,-1,0) and e3=(0,0,1,0,1)e_3=(0,0,1,0,-1). This is a basis of WW.


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