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This document provides instructions and questions for Test 1 of the Linear Algebra course for first year Bachelor of Technology students. The test contains 5 multiple part questions covering topics such as finding non-trivial solutions to systems of equations, defining vector space dimension, determining matrix rank and column space basis, finding a basis for the vector space of skew-symmetric matrices, evaluating matrix determinants, and determining linear independence of vectors. Students are instructed to show their work, write legibly, and not use programmable calculators or write on the question paper. The test is out of a maximum of 20 marks and has a duration of 1 hour.

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pradnya Ramgude
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394 views2 pages

La Paper Coep

This document provides instructions and questions for Test 1 of the Linear Algebra course for first year Bachelor of Technology students. The test contains 5 multiple part questions covering topics such as finding non-trivial solutions to systems of equations, defining vector space dimension, determining matrix rank and column space basis, finding a basis for the vector space of skew-symmetric matrices, evaluating matrix determinants, and determining linear independence of vectors. Students are instructed to show their work, write legibly, and not use programmable calculators or write on the question paper. The test is out of a maximum of 20 marks and has a duration of 1 hour.

Uploaded by

pradnya Ramgude
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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TEST 1 (SET B)

(MA15001 ) Linear Algebra


Course: F. Y. B.Tech , Semester I
Academic Year: 2016-2017 Max.Marks:20
Duration: 1 Hour Date: 11 /9/2016
Student MIS NO. :
Instructions:
(1) All questions are compulsory
(2) Figures to the right indicate maximum marks.
(3) Mobile phones and programmable calculators are strictly prohibited.
(4) Writing anything on question paper is not allowed.
(5) Write your MIS Number on Question Paper.

Q.1 i. Show that every n × n matrix is a sum of symmetric and skew-


symmetric matrix. [2]
ii. Find at least one non-trivial solution to x − 5y + 2z = 0. [2]

Q.2 i. Define dimension of a vector space V. [1]


T
ii. Explain with reasons whether the set S= {[x, y, z, w] | 2y −
3w = k; k ∈ R} is a subspace of R4 or not with respect to
usual operations. [3]

Q.3 Define the column rank of a matrix. Using row echelon form find
the rank and a basis for the column space of the matrix [4]
 
1 3 1 4
2 7 3 9
A=  1 5 3 1 .

1 2 0 8
1
2

P.T.O.

Q.4 Find a basis and hence the dimension of the vector space of all 3 × 3
skew-symmetric matrices with respect to usual operations. [4]

Q.5 i. Evaluate the determinant of the following matrix: [2]


 
1 2 0 3
 −1 0 0 1 
A=  1 −2 1 3 

0 0 −3 1
ii. If (a, b) and (c, d) be any two vectors in R2 . Show that if ad −
bc 6= 0 then (a, b) and (c, d) are linearly independent vectors in
R2 . [2]

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