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- For a 4×4 matrix whose top three rows are arbitrary and whose bottom row is (0, 0, 0, 1), show that the points (x, y, z, 1) and (hx, hy, hz, h) transform to the same point after homogenization.arrow_forwardBelow is the answers to problem 1 and 2, please help with problem 3. Problem 1 Use the svd() function in MATLAB to compute , the rank-1 approximation of . Clearly state what is, rounded to 4 decimal places. Also, compute the root-mean square error (RMSE) between and . Solution: %code %Define matrix A A = [1, 2, 3; 3, 3, 4; 5, 6, 7]; %Compute SVD of A [U, S, V] = svd(A); %Rank-1 approx A1 = U(:,1) * S(1,1) * V(:,1)'; RMSE = sqrt(mean((A(:) - A1(:)).^2)); %Display A1 rounded to 4 decimal places disp(round(A1, 4)); 1.7039 2.0313 2.4935 2.7243 3.2477 3.9867 4.9087 5.8517 7.1832 %Display RMSE disp(RMSE); 0.3257 Problem 2 Use the svd() function in MATLAB to compute , the rank-2 approximation of . Clearly state what is, rounded to 4 decimal places. Also, compute the root-mean square error (RMSE) between and . Which approximation is better, or ? Explain. Solution: %code A = [2, 4, 7; 3, 3, 5; 1, 6, 6]; % Compute SVD of A [U, S, V] = svd(A); % Rank-2…arrow_forwardFind a basis for and the dimension of the null space of the given matrix. (problem 32 on picture)arrow_forward
- Question1 Discuss the following terms with examples.invertible matrix, subspace, rank, column spacearrow_forwardONLY PARTS C AND D!!!! Given A and B are 3x3 matrices with det A = -2 and det B = 3 compute the following and show what property you are using directly in your work c) d et AT d) det B-1arrow_forwardThe reduced form R of a 3 by 3 matrix with randomly chosen entries is almost sure to be __ . What R is virtually certain if the random A is 4 by 3?arrow_forward
- Question 16: Classify the definiteness of the quadratic form associtated to the given matrix:arrow_forwardAn m×n matrix A is called upper triangular if all entries lying below the diagonal entries are zero, that is, if Aij= 0 whenever i > j. Prove that the upper triangular matrices form a subspace of Mm× n(F ).arrow_forwardSuppose A is 10 by 10 and A2 = 0 (zero matrix). So A multiplies each column of A to give the zero vector. This means that the column space of A is contained in the __ . If A has rank r, those subspaces have dimension r � 10 � r. So the rank is '(' � 5.arrow_forward
- In a matrix of cells, the cell traction force per unit mass is given by T ( n ) = an / 1+ bn 2 , where n is the number of cells, a is the measure of the traction force generated by a cell, and b measures how force is reduced due to neighboring cells. Find and interpret T ' ( n ) .arrow_forwardSuppose that S1 and S2 are subspaces of a vector space (V, F). Show that their intersection S1 ∩ S2 is also a subspace of (V, F). Is their union S1 ∪ S2 always a subspace?arrow_forwardWhat should k be for the matrix given in the question to be invertible?arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning