**Problem Statement:**Find a projection of a vector \[\begin{bmatrix}1 \\2 \\3 \\4\end{bmatrix}\]onto the subspace span\[\begin{pmatrix}\begin{bmatrix}2 \\5 \\4 \\5\end{bmatrix},\begin{bmatrix}1 \\1 \\2 \\7\end{bmatrix}\end{pmatrix}\]**Explanation:**The goal is to project the vector \(\begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix}\) onto the subspace spanned by the two given vectors \(\begin{bmatrix} 2 \\ 5 \\ 4 \\ 5 \end{bmatrix}\) and \(\begin{bmatrix} 1 \\ 1 \\ 2 \\ 7 \end{bmatrix}\). This involves determining the component of the vector that lies in the direction of the given subspace. The linear span indicates the combination of the basis vectors to form this subspace.

Formula to be used: The formula to calculate the projection of a vector v onto a non trivial subspace of a vector space V is given by: projSv=projv1v+projv2v+.......+projvrv=v·v1v1·v1v1+v·v2v2·v2v2+.......+v·vrvr·vrvr where, v1,v2, . . . . , vr form an orthogonal basis for the subspace S.Given: The vector v is given by: v=1234 The vectors v1 and v2 are given by: v1=2545 and v2=1127 To Find: We have to find a projection of the vector v onto the subspace span 2545 , 1127 .Solution: We have: v=1,2,3,4 , v1=2,5,4,5 and v2=1,1,2,7. The projection of vector v onto subspace is the vector given by: projSv=projv1v+projv2v projSv=v·v1v1·v1v1+v·v2v2·v2v2 projSv=1,2,3,4·2,5,4,52,5,4,5·2,5,4,52,5,4,5+1,2,3,4·1,1,2,71,1,2,7·1,1,2,71,1,2,7 We know that the dot product of two vectors a→=a1 i^+a2 j^+a3 k^ and b→=b1 i^+b2 j^+b3 k^ is given by: a·b=a1 i^+a2 j^+a3 k^·b1 i^+b2 j^+b3 k^=a1b1+a2b2+a3b3 Then, we have: projSv=12+25+34+4522+55+44+552,5,4,5+11+21+32+4711+11+22+771,1,2,7 projSv=2+10+12+204+25+16+252,5,4,5+1+2+6+281+1+4+491,1,2,7 projSv=44702,5,4,5+37551,1,2,7 projSv=22352,5,4,5+37551,1,2,7 projSv=4435,11035,8835,11035+3755,3755,7455,25955 projSv= 4435+3755 , 11035+3755 , 8835+7455 , 11035+25955 projSv= 1144+737385 , 11110+737385 , 1188+774385 , 11110+7259385 projSv= 484+259385 , 1210+259385 , 968+518385 , 1210+1813385 projSv= 743385 , 1469385 , 1486385 , 3023385 Hence, the projection of the vector v onto the given subspace is (743385,1469385,1486385,3023385) i.e., [743385146938514863853023385].