please can anyone help me to solve this 3 questions

question no {6,7 & 8}

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\fI. n- I. o. vae Theorem 2.22 {for the case of two subspaces 'R and 8 only}. '7. Let \"P" denote the vector space of polynomials of degree less than or equal to n, and of the form pix} = pg + pix + - - + + pnx", where the coefcients p; are all real. Let 'F'g denote the subspace of all even polynomials in \"P", i.e., those that satisfy the property p{x) = x). Similarly, let Pg denote the subspace of all odd polynomials1 i.e.. those satisfying P'Ix] : p{x]. Show mat 'P : \"PE EB 'Pg. 3. Repeat Example 2.23 using instead the two subspaces T of tritiiagonal matrices and Lt of upper triangular matrices. Example 2.20. 1. dimR") = n. 2. dimUR'm'\") 2 ma. Nate: Check that a basis fer Rm\" is given by the ma matees E;_,; E E E, j e where EU is a matrix all ef whose elements are Dexcept fora 1 in the (i, jJth local; The collection of E13,- matriees can be called the "natural basis matrices." 3+ dim(C[ro, r11) = +00. 4. dim{A 5 11am :A = AT} = %n(n +1). (To see why, determine 'I-HO'I + 1} symmtrie basis matrices.) 5. djmm E 1am : A is upper (lower) triangular} = nm +1). Thcurem 2.27. Far arbitrary subspaces R, 5 of a vector space V. dimf + S) = dimtfR) + dim{.5} dimtR 5L Example 2.23. Let It be the subspace at upper triangular matrices in R\" x\" and let L' be the subspace of lower triangular matrices in RM". Then it may be checked that E-I' + E. = R" "" while It [1 E is the set at diagnnaj matrices in 13"\". Using the fact that dim {diagonal matrices} = n, tngether with Examples 2.2[12 and 2.20.5. one can easin verify the validity cf the fnnnula given in Thenren] 2.2T