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Maths 2 week 3 important questions

Week 3 of Mathematics for Data Science II: Vector spaces, linear (in)dependence. Extra important questions based on the recent, changing patterns of the exams, from basic to advanced. Practise them in learning mode: each question has a worked solution.

Question 1

One or more correct options

Which of the following sets are subspaces of the corresponding vector space (of R3\mathbb R^3, R2\mathbb R^2, polynomials or matrices as indicated)? Select all.

Select all that apply.

  1. A

    {(x,y):x2=y2}\{(x,y): x^2=y^2\}

  2. B

    the set of all polynomials of degree at most 3

  3. C

    {(x,y):xy=0}\{(x,y): xy=0\}

  4. D

    the set of all invertible 2×22\times2 matrices

  5. E

    {(x,y,z):x+2y−z=0}\{(x,y,z): x+2y-z=0\}

  6. F

    the set of polynomials p(x)p(x) with p(1)=0p(1)=0

Question 2

One or more correct options

Select all the subspaces.

Select all that apply.

  1. A

    {(x,y,z):x+y+z≥0}\{(x,y,z): x+y+z\ge0\}

  2. B

    {(x,y):x2=y2}\{(x,y): x^2=y^2\}

  3. C

    {(x,y,z):x+y+z=1}\{(x,y,z): x+y+z=1\}

  4. D

    the set of all real sequences (a1,a2,a3)(a_1,a_2,a_3) with a1+a3=0a_1+a_3=0

  5. E

    the set of polynomials p(x)p(x) with p(1)=0p(1)=0

  6. F

    {(x,y,z):xyz=0}\{(x,y,z): xyz=0\}

Question 3

One or more correct options

Choose every set that is a subspace.

Select all that apply.

  1. A

    {(x,y,z):x+y=0 and y+z=0}\{(x,y,z): x+y=0\text{ and }y+z=0\}

  2. B

    {(x,y,z):x+y+z=1}\{(x,y,z): x+y+z=1\}

  3. C

    {(x,y,z):x=y=z}\{(x,y,z): x=y=z\}

  4. D

    the set of all symmetric 2×22\times2 matrices

  5. E

    the set of all vectors (x,y)(x,y) with ∣x∣≤1|x|\le1

  6. F

    the set of all polynomials of degree exactly 3

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